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          关于拉格朗日乘子法与KKT条件
         </a>
        </h1>
        <address class="msccaddress ">
         <em>
          12,880 次阅读 -
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      <div class="content-text">
       <p>
        作者：
        <a href="http://www.moozhi.com/user/show/54052c6a85b8c68f4ff19108">
         @
         <span class="username">
          wzyer
         </span>
        </a>
       </p>
       <p>
        拉格朗日乘子法无疑是最优化理论中最重要的一个方法。但是现在网上并没有很好的完整介绍整个方法的文章。我这里尝试详细介绍一下这方面的有关问题，插入自己的一些理解，希望能够对大家有帮助。本文分为两个部分：第一部分是数学上的定义以及公式上的推导；第二部分主要是一些常用方法的直观解释。初学者可以先看第二部分，但是第二部分会用到第一部分中的一些结论。请读者自行选择。
       </p>
       <h1>
        拉格朗日乘子法的数学基础
        <span id="anchor_JTI1dTYyQzklMjV1NjgzQyUyNXU2NzE3JTI1dTY1RTUlMjV1NEU1OCUyNXU1QjUwJTI1dTZDRDUlMjV1NzY4NCUyNXU2NTcwJTI1dTVCNjYlMjV1NTdGQSUyNXU3ODQwMA==">
        </span>
       </h1>
       <h2>
        共轭函数
        <span id="anchor_JTI1dTUxNzElMjV1OEY2RCUyNXU1MUZEJTI1dTY1NzAx">
        </span>
       </h2>
       <p>
        对于一个函数
        <span class="editor-math-element" data-id="JTI1MjRmJTI1M0ElMjUyMCUyNTdCJTI1NUNiZiUyNTIwUiUyNTdEJTI1NUVuJTI1MjAlMjU1Q3JpZ2h0YXJyb3clMjUyMCUyNTdCJTI1NUNiZiUyNTIwUiUyNTdEJTI1MjQ=">
         <span class="MathJax" id="MathJax-Element-1-Frame">
          <span class="math" id="MathJax-Span-1">
           <span class="mrow" id="MathJax-Span-2">
            <span class="mi" id="MathJax-Span-3">
             f
            </span>
            <span class="mo" id="MathJax-Span-4">
             :
            </span>
            <span class="msubsup" id="MathJax-Span-5">
             <span class="texatom" id="MathJax-Span-6">
              <span class="mrow" id="MathJax-Span-7">
               <span class="mi" id="MathJax-Span-8">
                R
               </span>
              </span>
             </span>
             <span class="mi" id="MathJax-Span-9">
              n
             </span>
            </span>
            <span class="mo" id="MathJax-Span-10">
             →
            </span>
            <span class="texatom" id="MathJax-Span-11">
             <span class="mrow" id="MathJax-Span-12">
              <span class="mi" id="MathJax-Span-13">
               R
              </span>
             </span>
            </span>
           </span>
          </span>
         </span>
        </span>
        （不要求是凸函数），我们可以定义它的共轭函数
        <span class="editor-math-element" data-id="JTI1MjRmJTI1NUUlMjU1Q3N0YXIlMjUzQSUyNTdCJTI1NUNiZiUyNTIwUiUyNTdEJTI1NUVuJTI1MjAlMjU1Q3JpZ2h0YXJyb3clMjUyMCUyNTdCJTI1NUNiZiUyNTIwUiUyNTdEJTI1MjQ=">
         <span class="MathJax" id="MathJax-Element-2-Frame">
          <span class="math" id="MathJax-Span-14">
           <span class="mrow" id="MathJax-Span-15">
            <span class="msubsup" id="MathJax-Span-16">
             <span class="mi" id="MathJax-Span-17">
              f
             </span>
             <span class="mo" id="MathJax-Span-18">
              ⋆
             </span>
            </span>
            <span class="mo" id="MathJax-Span-19">
             :
            </span>
            <span class="msubsup" id="MathJax-Span-20">
             <span class="texatom" id="MathJax-Span-21">
              <span class="mrow" id="MathJax-Span-22">
               <span class="mi" id="MathJax-Span-23">
                R
               </span>
              </span>
             </span>
             <span class="mi" id="MathJax-Span-24">
              n
             </span>
            </span>
            <span class="mo" id="MathJax-Span-25">
             →
            </span>
            <span class="texatom" id="MathJax-Span-26">
             <span class="mrow" id="MathJax-Span-27">
              <span class="mi" id="MathJax-Span-28">
               R
              </span>
             </span>
            </span>
           </span>
          </span>
         </span>
        </span>
        为：
       </p>
       <div class="MathJax_Display">
        <span class="MathJax" id="MathJax-Element-3-Frame">
         <span class="math" id="MathJax-Span-29">
          <span class="mrow" id="MathJax-Span-30">
           <span class="msubsup" id="MathJax-Span-31">
            <span class="mi" id="MathJax-Span-32">
             f
            </span>
            <span class="mo" id="MathJax-Span-33">
             ⋆
            </span>
           </span>
           <span class="mo" id="MathJax-Span-34">
            (
           </span>
           <span class="texatom" id="MathJax-Span-35">
            <span class="mrow" id="MathJax-Span-36">
             <span class="mi" id="MathJax-Span-37">
              y
             </span>
            </span>
           </span>
           <span class="mo" id="MathJax-Span-38">
            )
           </span>
           <span class="mo" id="MathJax-Span-39">
            =
           </span>
           <span class="munderover" id="MathJax-Span-40">
            <span class="mo" id="MathJax-Span-41">
             sup
            </span>
            <span class="texatom" id="MathJax-Span-42">
             <span class="mrow" id="MathJax-Span-43">
              <span class="mi" id="MathJax-Span-44">
               x
              </span>
              <span class="mo" id="MathJax-Span-45">
               ∈
              </span>
              <span class="mtext" id="MathJax-Span-46">
               dom
              </span>
              <span class="mi" id="MathJax-Span-47">
               f
              </span>
             </span>
            </span>
           </span>
           <span class="mo" id="MathJax-Span-48">
            (
           </span>
           <span class="msubsup" id="MathJax-Span-49">
            <span class="texatom" id="MathJax-Span-50">
             <span class="mrow" id="MathJax-Span-51">
              <span class="mi" id="MathJax-Span-52">
               y
              </span>
             </span>
            </span>
            <span class="mi" id="MathJax-Span-53">
             T
            </span>
           </span>
           <span class="texatom" id="MathJax-Span-54">
            <span class="mrow" id="MathJax-Span-55">
             <span class="mi" id="MathJax-Span-56">
              x
             </span>
            </span>
           </span>
           <span class="mo" id="MathJax-Span-57">
            −
           </span>
           <span class="mi" id="MathJax-Span-58">
            f
           </span>
           <span class="mo" id="MathJax-Span-59">
            (
           </span>
           <span class="texatom" id="MathJax-Span-60">
            <span class="mrow" id="MathJax-Span-61">
             <span class="mi" id="MathJax-Span-62">
              x
             </span>
            </span>
           </span>
           <span class="mo" id="MathJax-Span-63">
            )
           </span>
           <span class="mo" id="MathJax-Span-64">
            )
           </span>
           <span class="mtext" id="MathJax-Span-65">
           </span>
          </span>
         </span>
        </span>
       </div>
       <p>
        按照上面的定义，这个函数值是有可能取到
        <span class="editor-math-element" data-id="JTI1MjQlMjU1Q2luZnR5JTI1MjQ=">
         <span class="MathJax" id="MathJax-Element-4-Frame">
          <span class="math" id="MathJax-Span-66">
           <span class="mrow" id="MathJax-Span-67">
            <span class="mi" id="MathJax-Span-68">
             ∞
            </span>
           </span>
          </span>
         </span>
        </span>
        的，这并不是我们希望的结果。因此我们规定，使函数值有界的
        <span class="editor-math-element" data-id="JTI1MjR5JTI1MjQ=">
         <span class="MathJax" id="MathJax-Element-5-Frame">
          <span class="math" id="MathJax-Span-69">
           <span class="mrow" id="MathJax-Span-70">
            <span class="mi" id="MathJax-Span-71">
             y
            </span>
           </span>
          </span>
         </span>
        </span>
        的取值范围就是共轭函数的定义域。
       </p>
       <p>
        下面是一些常用函数的对偶函数：
       </p>
       <ul>
        <li>
         线性函数
         <span class="editor-math-element" data-id="JTI1MjRmJTI1Mjh4JTI1MjklMjUzRGF4JTJCYiUyNTI0">
          <span class="MathJax" id="MathJax-Element-6-Frame">
           <span class="math" id="MathJax-Span-72">
            <span class="mrow" id="MathJax-Span-73">
             <span class="mi" id="MathJax-Span-74">
              f
             </span>
             <span class="mo" id="MathJax-Span-75">
              (
             </span>
             <span class="mi" id="MathJax-Span-76">
              x
             </span>
             <span class="mo" id="MathJax-Span-77">
              )
             </span>
             <span class="mo" id="MathJax-Span-78">
              =
             </span>
             <span class="mi" id="MathJax-Span-79">
              a
             </span>
             <span class="mi" id="MathJax-Span-80">
              x
             </span>
             <span class="mo" id="MathJax-Span-81">
              +
             </span>
             <span class="mi" id="MathJax-Span-82">
              b
             </span>
            </span>
           </span>
          </span>
         </span>
         的共轭函数是
         <span class="editor-math-element" data-id="JTI1MjRmJTI1NUUlMjU1Q2FzdCUyNTI4YSUyNTI5JTI1M0QtYiUyNTI0">
          <span class="MathJax" id="MathJax-Element-7-Frame">
           <span class="math" id="MathJax-Span-83">
            <span class="mrow" id="MathJax-Span-84">
             <span class="msubsup" id="MathJax-Span-85">
              <span class="mi" id="MathJax-Span-86">
               f
              </span>
              <span class="mo" id="MathJax-Span-87">
               ∗
              </span>
             </span>
             <span class="mo" id="MathJax-Span-88">
              (
             </span>
             <span class="mi" id="MathJax-Span-89">
              a
             </span>
             <span class="mo" id="MathJax-Span-90">
              )
             </span>
             <span class="mo" id="MathJax-Span-91">
              =
             </span>
             <span class="mo" id="MathJax-Span-92">
              −
             </span>
             <span class="mi" id="MathJax-Span-93">
              b
             </span>
            </span>
           </span>
          </span>
         </span>
         ，定义域是单点
         <span class="editor-math-element" data-id="JTI1MjQlMjU3QmElMjU3RCUyNTI0">
          <span class="MathJax" id="MathJax-Element-8-Frame">
           <span class="math" id="MathJax-Span-94">
            <span class="mrow" id="MathJax-Span-95">
             <span class="texatom" id="MathJax-Span-96">
              <span class="mrow" id="MathJax-Span-97">
               <span class="mi" id="MathJax-Span-98">
                a
               </span>
              </span>
             </span>
            </span>
           </span>
          </span>
         </span>
         。
        </li>
        <li>
         负对数
         <span class="editor-math-element" data-id="JTI1MjRmJTI1Mjh4JTI1MjklMjUzRC0lMjU1Q2xvZyUyNTdCeCUyNTdEJTI1MjQ=">
          <span class="MathJax" id="MathJax-Element-9-Frame">
           <span class="math" id="MathJax-Span-99">
            <span class="mrow" id="MathJax-Span-100">
             <span class="mi" id="MathJax-Span-101">
              f
             </span>
             <span class="mo" id="MathJax-Span-102">
              (
             </span>
             <span class="mi" id="MathJax-Span-103">
              x
             </span>
             <span class="mo" id="MathJax-Span-104">
              )
             </span>
             <span class="mo" id="MathJax-Span-105">
              =
             </span>
             <span class="mo" id="MathJax-Span-106">
              −
             </span>
             <span class="mi" id="MathJax-Span-107">
              log
             </span>
             <span class="texatom" id="MathJax-Span-108">
              <span class="mrow" id="MathJax-Span-109">
               <span class="mi" id="MathJax-Span-110">
                x
               </span>
              </span>
             </span>
            </span>
           </span>
          </span>
         </span>
         的共轭函数是
         <span class="editor-math-element" data-id="JTI1MjRmJTI1NUUlMjU1Q2FzdCUyNTI4eSUyNTI5JTI1M0QtJTI1NUNsb2clMjU3QiUyNTI4LXklMjUyOSUyNTdELTElMjUyNA==">
          <span class="MathJax" id="MathJax-Element-10-Frame">
           <span class="math" id="MathJax-Span-111">
            <span class="mrow" id="MathJax-Span-112">
             <span class="msubsup" id="MathJax-Span-113">
              <span class="mi" id="MathJax-Span-114">
               f
              </span>
              <span class="mo" id="MathJax-Span-115">
               ∗
              </span>
             </span>
             <span class="mo" id="MathJax-Span-116">
              (
             </span>
             <span class="mi" id="MathJax-Span-117">
              y
             </span>
             <span class="mo" id="MathJax-Span-118">
              )
             </span>
             <span class="mo" id="MathJax-Span-119">
              =
             </span>
             <span class="mo" id="MathJax-Span-120">
              −
             </span>
             <span class="mi" id="MathJax-Span-121">
              log
             </span>
             <span class="texatom" id="MathJax-Span-122">
              <span class="mrow" id="MathJax-Span-123">
               <span class="mo" id="MathJax-Span-124">
                (
               </span>
               <span class="mo" id="MathJax-Span-125">
                −
               </span>
               <span class="mi" id="MathJax-Span-126">
                y
               </span>
               <span class="mo" id="MathJax-Span-127">
                )
               </span>
              </span>
             </span>
             <span class="mo" id="MathJax-Span-128">
              −
             </span>
             <span class="mn" id="MathJax-Span-129">
              1
             </span>
            </span>
           </span>
          </span>
         </span>
         ，定义域是
         <span class="editor-math-element" data-id="JTI1MjR5JTI1MjZsdCUyNTNCMCUyNTI0">
          <span class="MathJax" id="MathJax-Element-11-Frame">
           <span class="math" id="MathJax-Span-130">
            <span class="mrow" id="MathJax-Span-131">
             <span class="mi" id="MathJax-Span-132">
              y
             </span>
             <span class="mo" id="MathJax-Span-133">
              &lt;
             </span>
             <span class="mn" id="MathJax-Span-134">
              0
             </span>
            </span>
           </span>
          </span>
         </span>
         。
        </li>
        <li>
         指数函数
         <span class="editor-math-element" data-id="JTI1MjRmJTI1Mjh4JTI1MjklMjUzRCUyNTVDbWF0aHJtJTI1N0JlJTI1N0QlMjU1RXglMjUyNA==">
          <span class="MathJax" id="MathJax-Element-12-Frame">
           <span class="math" id="MathJax-Span-135">
            <span class="mrow" id="MathJax-Span-136">
             <span class="mi" id="MathJax-Span-137">
              f
             </span>
             <span class="mo" id="MathJax-Span-138">
              (
             </span>
             <span class="mi" id="MathJax-Span-139">
              x
             </span>
             <span class="mo" id="MathJax-Span-140">
              )
             </span>
             <span class="mo" id="MathJax-Span-141">
              =
             </span>
             <span class="msubsup" id="MathJax-Span-142">
              <span class="texatom" id="MathJax-Span-143">
               <span class="mrow" id="MathJax-Span-144">
                <span class="mi" id="MathJax-Span-145">
                 e
                </span>
               </span>
              </span>
              <span class="mi" id="MathJax-Span-146">
               x
              </span>
             </span>
            </span>
           </span>
          </span>
         </span>
         的共轭函数是
         <span class="editor-math-element" data-id="JTI1MjRmJTI1NUUlMjU1Q2FzdCUyNTI4eSUyNTI5JTI1M0R5JTI1NUNsb2clMjU3QnklMjU3RC15JTI1MjQ=">
          <span class="MathJax" id="MathJax-Element-13-Frame">
           <span class="math" id="MathJax-Span-147">
            <span class="mrow" id="MathJax-Span-148">
             <span class="msubsup" id="MathJax-Span-149">
              <span class="mi" id="MathJax-Span-150">
               f
              </span>
              <span class="mo" id="MathJax-Span-151">
               ∗
              </span>
             </span>
             <span class="mo" id="MathJax-Span-152">
              (
             </span>
             <span class="mi" id="MathJax-Span-153">
              y
             </span>
             <span class="mo" id="MathJax-Span-154">
              )
             </span>
             <span class="mo" id="MathJax-Span-155">
              =
             </span>
             <span class="mi" id="MathJax-Span-156">
              y
             </span>
             <span class="mi" id="MathJax-Span-157">
              log
             </span>
             <span class="texatom" id="MathJax-Span-158">
              <span class="mrow" id="MathJax-Span-159">
               <span class="mi" id="MathJax-Span-160">
                y
               </span>
              </span>
             </span>
             <span class="mo" id="MathJax-Span-161">
              −
             </span>
             <span class="mi" id="MathJax-Span-162">
              y
             </span>
            </span>
           </span>
          </span>
         </span>
         ，定义域是
         <span class="editor-math-element" data-id="JTI1MjR5JTI1MjAlMjU1Q2dlJTI1MjAwJTI1MjQ=">
          <span class="MathJax" id="MathJax-Element-14-Frame">
           <span class="math" id="MathJax-Span-163">
            <span class="mrow" id="MathJax-Span-164">
             <span class="mi" id="MathJax-Span-165">
              y
             </span>
             <span class="mo" id="MathJax-Span-166">
              ≥
             </span>
             <span class="mn" id="MathJax-Span-167">
              0
             </span>
            </span>
           </span>
          </span>
         </span>
         。
        </li>
        <li>
         负熵函数
         <span class="editor-math-element" data-id="JTI1MjRmJTI1Mjh4JTI1MjklMjUzRHglMjU1Q2xvZyUyNTdCeCUyNTdEJTI1MjQ=">
          <span class="MathJax" id="MathJax-Element-15-Frame">
           <span class="math" id="MathJax-Span-168">
            <span class="mrow" id="MathJax-Span-169">
             <span class="mi" id="MathJax-Span-170">
              f
             </span>
             <span class="mo" id="MathJax-Span-171">
              (
             </span>
             <span class="mi" id="MathJax-Span-172">
              x
             </span>
             <span class="mo" id="MathJax-Span-173">
              )
             </span>
             <span class="mo" id="MathJax-Span-174">
              =
             </span>
             <span class="mi" id="MathJax-Span-175">
              x
             </span>
             <span class="mi" id="MathJax-Span-176">
              log
             </span>
             <span class="texatom" id="MathJax-Span-177">
              <span class="mrow" id="MathJax-Span-178">
               <span class="mi" id="MathJax-Span-179">
                x
               </span>
              </span>
             </span>
            </span>
           </span>
          </span>
         </span>
         的共轭函数是
         <span class="editor-math-element" data-id="JTI1MjRmJTI1NUUlMjU1Q2FzdCUyNTI4eSUyNTI5JTI1M0QlMjU1Q21hdGhybSUyNTdCZSUyNTdEJTI1NUUlMjU3QnktMSUyNTdEJTI1MjQ=">
          <span class="MathJax" id="MathJax-Element-16-Frame">
           <span class="math" id="MathJax-Span-180">
            <span class="mrow" id="MathJax-Span-181">
             <span class="msubsup" id="MathJax-Span-182">
              <span class="mi" id="MathJax-Span-183">
               f
              </span>
              <span class="mo" id="MathJax-Span-184">
               ∗
              </span>
             </span>
             <span class="mo" id="MathJax-Span-185">
              (
             </span>
             <span class="mi" id="MathJax-Span-186">
              y
             </span>
             <span class="mo" id="MathJax-Span-187">
              )
             </span>
             <span class="mo" id="MathJax-Span-188">
              =
             </span>
             <span class="msubsup" id="MathJax-Span-189">
              <span class="texatom" id="MathJax-Span-190">
               <span class="mrow" id="MathJax-Span-191">
                <span class="mi" id="MathJax-Span-192">
                 e
                </span>
               </span>
              </span>
              <span class="texatom" id="MathJax-Span-193">
               <span class="mrow" id="MathJax-Span-194">
                <span class="mi" id="MathJax-Span-195">
                 y
                </span>
                <span class="mo" id="MathJax-Span-196">
                 −
                </span>
                <span class="mn" id="MathJax-Span-197">
                 1
                </span>
               </span>
              </span>
             </span>
            </span>
           </span>
          </span>
         </span>
         ，定义域是
         <span class="editor-math-element" data-id="JTI1MjQlMjU1Q2JmJTI1MjBSJTI1MjQ=">
          <span class="MathJax" id="MathJax-Element-17-Frame">
           <span class="math" id="MathJax-Span-198">
            <span class="mrow" id="MathJax-Span-199">
             <span class="mi" id="MathJax-Span-200">
              R
             </span>
            </span>
           </span>
          </span>
         </span>
        </li>
        <li>
         倒数函数
         <span class="editor-math-element" data-id="JTI1MjRmJTI1Mjh4JTI1MjklMjUzRDElMkZ4JTI1MjQ=">
          <span class="MathJax" id="MathJax-Element-18-Frame">
           <span class="math" id="MathJax-Span-201">
            <span class="mrow" id="MathJax-Span-202">
             <span class="mi" id="MathJax-Span-203">
              f
             </span>
             <span class="mo" id="MathJax-Span-204">
              (
             </span>
             <span class="mi" id="MathJax-Span-205">
              x
             </span>
             <span class="mo" id="MathJax-Span-206">
              )
             </span>
             <span class="mo" id="MathJax-Span-207">
              =
             </span>
             <span class="mn" id="MathJax-Span-208">
              1
             </span>
             <span class="texatom" id="MathJax-Span-209">
              <span class="mrow" id="MathJax-Span-210">
               <span class="mo" id="MathJax-Span-211">
                /
               </span>
              </span>
             </span>
             <span class="mi" id="MathJax-Span-212">
              x
             </span>
            </span>
           </span>
          </span>
         </span>
         的共轭函数是
         <span class="editor-math-element" data-id="JTI1MjRmJTI1NUUlMjU1Q2FzdCUyNTI4eSUyNTI5JTI1M0QtMiUyNTI4LXklMjUyOSUyNTVFJTI1N0IxJTJGMiUyNTdEJTI1MjQ=">
          <span class="MathJax" id="MathJax-Element-19-Frame">
           <span class="math" id="MathJax-Span-213">
            <span class="mrow" id="MathJax-Span-214">
             <span class="msubsup" id="MathJax-Span-215">
              <span class="mi" id="MathJax-Span-216">
               f
              </span>
              <span class="mo" id="MathJax-Span-217">
               ∗
              </span>
             </span>
             <span class="mo" id="MathJax-Span-218">
              (
             </span>
             <span class="mi" id="MathJax-Span-219">
              y
             </span>
             <span class="mo" id="MathJax-Span-220">
              )
             </span>
             <span class="mo" id="MathJax-Span-221">
              =
             </span>
             <span class="mo" id="MathJax-Span-222">
              −
             </span>
             <span class="mn" id="MathJax-Span-223">
              2
             </span>
             <span class="mo" id="MathJax-Span-224">
              (
             </span>
             <span class="mo" id="MathJax-Span-225">
              −
             </span>
             <span class="mi" id="MathJax-Span-226">
              y
             </span>
             <span class="msubsup" id="MathJax-Span-227">
              <span class="mo" id="MathJax-Span-228">
               )
              </span>
              <span class="texatom" id="MathJax-Span-229">
               <span class="mrow" id="MathJax-Span-230">
                <span class="mn" id="MathJax-Span-231">
                 1
                </span>
                <span class="texatom" id="MathJax-Span-232">
                 <span class="mrow" id="MathJax-Span-233">
                  <span class="mo" id="MathJax-Span-234">
                   /
                  </span>
                 </span>
                </span>
                <span class="mn" id="MathJax-Span-235">
                 2
                </span>
               </span>
              </span>
             </span>
            </span>
           </span>
          </span>
         </span>
         ，定义域数
         <span class="editor-math-element" data-id="JTI1MjR5JTI1NUNsZTAlMjUyNA==">
          <span class="MathJax" id="MathJax-Element-20-Frame">
           <span class="math" id="MathJax-Span-236">
            <span class="mrow" id="MathJax-Span-237">
             <span class="mi" id="MathJax-Span-238">
              y
             </span>
             <span class="mo" id="MathJax-Span-239">
              ≤
             </span>
             <span class="mn" id="MathJax-Span-240">
              0
             </span>
            </span>
           </span>
          </span>
         </span>
         。
        </li>
        <li>
         任意范数
         <span class="editor-math-element" data-id="JTI1MjRmJTI1Mjh4JTI1MjklMjUzRCUyNTVDVmVydCUyNTIweCUyNTIwJTI1NUNWZXJ0JTI1MjQ=">
          <span class="MathJax" id="MathJax-Element-21-Frame">
           <span class="math" id="MathJax-Span-241">
            <span class="mrow" id="MathJax-Span-242">
             <span class="mi" id="MathJax-Span-243">
              f
             </span>
             <span class="mo" id="MathJax-Span-244">
              (
             </span>
             <span class="mi" id="MathJax-Span-245">
              x
             </span>
             <span class="mo" id="MathJax-Span-246">
              )
             </span>
             <span class="mo" id="MathJax-Span-247">
              =
             </span>
             <span class="mo" id="MathJax-Span-248">
              ∥
             </span>
             <span class="mi" id="MathJax-Span-249">
              x
             </span>
             <span class="mo" id="MathJax-Span-250">
              ∥
             </span>
            </span>
           </span>
          </span>
         </span>
         的共轭函数是
         <br/>
         <span class="editor-math-element" data-id="JTI1MjRmJTI1NUUlMjU1Q2FzdCUyNTI4eSUyNTI5JTI1M0QlMjU1Q2JlZ2luJTI1N0JjYXNlcyUyNTdEJTI1MjAwJTI1MjAlMjUyNiUyNTIwJTI1NUNWZXJ0JTI1MjB5JTI1MjAlMjU1Q1ZlcnRfJTI1NUNhc3QlMjUyMCUyNTVDbGVxJTI1MjAxJTI1MjAlMjU1QyUyNTVDJTI1MjAlMjU1Q2luZnR5JTI1MjAlMjUyNiUyNTIwJTI1NUN0ZXh0JTI1N0JvdGhlcndpc2UlMjU3RCUyNTIwJTI1NUNlbmQlMjU3QmNhc2VzJTI1N0QlMjUyNA==">
          <span class="MathJax" id="MathJax-Element-22-Frame">
           <span class="math" id="MathJax-Span-251">
            <span class="mrow" id="MathJax-Span-252">
             <span class="msubsup" id="MathJax-Span-253">
              <span class="mi" id="MathJax-Span-254">
               f
              </span>
              <span class="mo" id="MathJax-Span-255">
               ∗
              </span>
             </span>
             <span class="mo" id="MathJax-Span-256">
              (
             </span>
             <span class="mi" id="MathJax-Span-257">
              y
             </span>
             <span class="mo" id="MathJax-Span-258">
              )
             </span>
             <span class="mo" id="MathJax-Span-259">
              =
             </span>
             <span class="mrow" id="MathJax-Span-260">
              <span class="mo" id="MathJax-Span-261">
               {
              </span>
              <span class="mtable" id="MathJax-Span-262">
               <span class="mtd" id="MathJax-Span-263">
                <span class="mrow" id="MathJax-Span-264">
                 <span class="mn" id="MathJax-Span-265">
                  0
                 </span>
                </span>
               </span>
               <span class="mtd" id="MathJax-Span-275">
                <span class="mrow" id="MathJax-Span-276">
                 <span class="mi" id="MathJax-Span-277">
                  ∞
                 </span>
                </span>
               </span>
               <span class="mtd" id="MathJax-Span-266">
                <span class="mrow" id="MathJax-Span-267">
                 <span class="mo" id="MathJax-Span-268">
                  ∥
                 </span>
                 <span class="mi" id="MathJax-Span-269">
                  y
                 </span>
                 <span class="msubsup" id="MathJax-Span-270">
                  <span class="mo" id="MathJax-Span-271">
                   ∥
                  </span>
                  <span class="mo" id="MathJax-Span-272">
                   ∗
                  </span>
                 </span>
                 <span class="mo" id="MathJax-Span-273">
                  ≤
                 </span>
                 <span class="mn" id="MathJax-Span-274">
                  1
                 </span>
                </span>
               </span>
               <span class="mtd" id="MathJax-Span-278">
                <span class="mrow" id="MathJax-Span-279">
                 <span class="mtext" id="MathJax-Span-280">
                  otherwise
                 </span>
                </span>
               </span>
              </span>
             </span>
            </span>
           </span>
          </span>
         </span>
         <br/>
         其中
         <span class="editor-math-element" data-id="JTI1MjQlMjU1Q1ZlcnQlMjUyMHklMjUyMCUyNTVDVmVydF8lMjU1Q2FzdCUyNTI0">
          <span class="MathJax" id="MathJax-Element-23-Frame">
           <span class="math" id="MathJax-Span-281">
            <span class="mrow" id="MathJax-Span-282">
             <span class="mo" id="MathJax-Span-283">
              ∥
             </span>
             <span class="mi" id="MathJax-Span-284">
              y
             </span>
             <span class="msubsup" id="MathJax-Span-285">
              <span class="mo" id="MathJax-Span-286">
               ∥
              </span>
              <span class="mo" id="MathJax-Span-287">
               ∗
              </span>
             </span>
            </span>
           </span>
          </span>
         </span>
         是原范数
         <span class="editor-math-element" data-id="JTI1MjQlMjU1Q1ZlcnQlMjUyMHglMjUyMCUyNTVDVmVydCUyNTI0">
          <span class="MathJax" id="MathJax-Element-24-Frame">
           <span class="math" id="MathJax-Span-288">
            <span class="mrow" id="MathJax-Span-289">
             <span class="mo" id="MathJax-Span-290">
              ∥
             </span>
             <span class="mi" id="MathJax-Span-291">
              x
             </span>
             <span class="mo" id="MathJax-Span-292">
              ∥
             </span>
            </span>
           </span>
          </span>
         </span>
         的对偶范数。
        </li>
        <li>
         范数平方
         <span class="editor-math-element" data-id="JTI1MjRmJTI1Mjh4JTI1MjklMjUzRCUyNTI4MSUyRjIlMjUyOSUyNTVDVmVydCUyNTIweCUyNTIwJTI1NUNWZXJ0JTI1NUUyJTI1MjQ=">
          <span class="MathJax" id="MathJax-Element-25-Frame">
           <span class="math" id="MathJax-Span-293">
            <span class="mrow" id="MathJax-Span-294">
             <span class="mi" id="MathJax-Span-295">
              f
             </span>
             <span class="mo" id="MathJax-Span-296">
              (
             </span>
             <span class="mi" id="MathJax-Span-297">
              x
             </span>
             <span class="mo" id="MathJax-Span-298">
              )
             </span>
             <span class="mo" id="MathJax-Span-299">
              =
             </span>
             <span class="mo" id="MathJax-Span-300">
              (
             </span>
             <span class="mn" id="MathJax-Span-301">
              1
             </span>
             <span class="texatom" id="MathJax-Span-302">
              <span class="mrow" id="MathJax-Span-303">
               <span class="mo" id="MathJax-Span-304">
                /
               </span>
              </span>
             </span>
             <span class="mn" id="MathJax-Span-305">
              2
             </span>
             <span class="mo" id="MathJax-Span-306">
              )
             </span>
             <span class="mo" id="MathJax-Span-307">
              ∥
             </span>
             <span class="mi" id="MathJax-Span-308">
              x
             </span>
             <span class="msubsup" id="MathJax-Span-309">
              <span class="mo" id="MathJax-Span-310">
               ∥
              </span>
              <span class="mn" id="MathJax-Span-311">
               2
              </span>
             </span>
            </span>
           </span>
          </span>
         </span>
         的共轭函数是
         <span class="editor-math-element" data-id="JTI1MjRmJTI1NUUlMjU1Q2FzdCUyNTI4eSUyNTI5JTI1M0QlMjUyODElMkYyJTI1MjklMjU1Q1ZlcnQlMjUyMHklMjUyMCUyNTVDVmVydF8lMjU1Q2FzdCUyNTVFMiUyNTI0">
          <span class="MathJax" id="MathJax-Element-26-Frame">
           <span class="math" id="MathJax-Span-312">
            <span class="mrow" id="MathJax-Span-313">
             <span class="msubsup" id="MathJax-Span-314">
              <span class="mi" id="MathJax-Span-315">
               f
              </span>
              <span class="mo" id="MathJax-Span-316">
               ∗
              </span>
             </span>
             <span class="mo" id="MathJax-Span-317">
              (
             </span>
             <span class="mi" id="MathJax-Span-318">
              y
             </span>
             <span class="mo" id="MathJax-Span-319">
              )
             </span>
             <span class="mo" id="MathJax-Span-320">
              =
             </span>
             <span class="mo" id="MathJax-Span-321">
              (
             </span>
             <span class="mn" id="MathJax-Span-322">
              1
             </span>
             <span class="texatom" id="MathJax-Span-323">
              <span class="mrow" id="MathJax-Span-324">
               <span class="mo" id="MathJax-Span-325">
                /
               </span>
              </span>
             </span>
             <span class="mn" id="MathJax-Span-326">
              2
             </span>
             <span class="mo" id="MathJax-Span-327">
              )
             </span>
             <span class="mo" id="MathJax-Span-328">
              ∥
             </span>
             <span class="mi" id="MathJax-Span-329">
              y
             </span>
             <span class="msubsup" id="MathJax-Span-330">
              <span class="mo" id="MathJax-Span-331">
               ∥
              </span>
              <span class="mn" id="MathJax-Span-332">
               2
              </span>
              <span class="mo" id="MathJax-Span-333">
               ∗
              </span>
             </span>
            </span>
           </span>
          </span>
         </span>
         ，其中
         <span class="editor-math-element" data-id="JTI1MjQlMjU1Q1ZlcnQlMjUyMHklMjUyMCUyNTVDVmVydF8lMjU1Q2FzdCUyNTI0">
          <span class="MathJax" id="MathJax-Element-27-Frame">
           <span class="math" id="MathJax-Span-334">
            <span class="mrow" id="MathJax-Span-335">
             <span class="mo" id="MathJax-Span-336">
              ∥
             </span>
             <span class="mi" id="MathJax-Span-337">
              y
             </span>
             <span class="msubsup" id="MathJax-Span-338">
              <span class="mo" id="MathJax-Span-339">
               ∥
              </span>
              <span class="mo" id="MathJax-Span-340">
               ∗
              </span>
             </span>
            </span>
           </span>
          </span>
         </span>
         是原范数
         <span class="editor-math-element" data-id="JTI1MjQlMjU1Q1ZlcnQlMjUyMHglMjUyMCUyNTVDVmVydCUyNTI0">
          <span class="MathJax" id="MathJax-Element-28-Frame">
           <span class="math" id="MathJax-Span-341">
            <span class="mrow" id="MathJax-Span-342">
             <span class="mo" id="MathJax-Span-343">
              ∥
             </span>
             <span class="mi" id="MathJax-Span-344">
              x
             </span>
             <span class="mo" id="MathJax-Span-345">
              ∥
             </span>
            </span>
           </span>
          </span>
         </span>
         的对偶范数。
        </li>
        <li>
         二次型
         <span class="editor-math-element" data-id="JTI1MjRmJTI1Mjh4JTI1MjklMjUzRCUyNTI4MSUyRjIlMjUyOXglMjU1RVQlMjUyMFElMjUyMHglMjUyNA==">
          <span class="MathJax" id="MathJax-Element-29-Frame">
           <span class="math" id="MathJax-Span-346">
            <span class="mrow" id="MathJax-Span-347">
             <span class="mi" id="MathJax-Span-348">
              f
             </span>
             <span class="mo" id="MathJax-Span-349">
              (
             </span>
             <span class="mi" id="MathJax-Span-350">
              x
             </span>
             <span class="mo" id="MathJax-Span-351">
              )
             </span>
             <span class="mo" id="MathJax-Span-352">
              =
             </span>
             <span class="mo" id="MathJax-Span-353">
              (
             </span>
             <span class="mn" id="MathJax-Span-354">
              1
             </span>
             <span class="texatom" id="MathJax-Span-355">
              <span class="mrow" id="MathJax-Span-356">
               <span class="mo" id="MathJax-Span-357">
                /
               </span>
              </span>
             </span>
             <span class="mn" id="MathJax-Span-358">
              2
             </span>
             <span class="mo" id="MathJax-Span-359">
              )
             </span>
             <span class="msubsup" id="MathJax-Span-360">
              <span class="mi" id="MathJax-Span-361">
               x
              </span>
              <span class="mi" id="MathJax-Span-362">
               T
              </span>
             </span>
             <span class="mi" id="MathJax-Span-363">
              Q
             </span>
             <span class="mi" id="MathJax-Span-364">
              x
             </span>
            </span>
           </span>
          </span>
         </span>
         的共轭函数是
         <span class="editor-math-element" data-id="JTI1MjRmJTI1NUUlMjU1Q2FzdCUyNTI4eSUyNTI5JTI1M0QlMjUyODElMkYyJTI1Mjl5JTI1NUVUJTI1MjBRJTI1NUUlMjU3Qi0xJTI1N0QlMjUyMHklMjUyNA==">
          <span class="MathJax" id="MathJax-Element-30-Frame">
           <span class="math" id="MathJax-Span-365">
            <span class="mrow" id="MathJax-Span-366">
             <span class="msubsup" id="MathJax-Span-367">
              <span class="mi" id="MathJax-Span-368">
               f
              </span>
              <span class="mo" id="MathJax-Span-369">
               ∗
              </span>
             </span>
             <span class="mo" id="MathJax-Span-370">
              (
             </span>
             <span class="mi" id="MathJax-Span-371">
              y
             </span>
             <span class="mo" id="MathJax-Span-372">
              )
             </span>
             <span class="mo" id="MathJax-Span-373">
              =
             </span>
             <span class="mo" id="MathJax-Span-374">
              (
             </span>
             <span class="mn" id="MathJax-Span-375">
              1
             </span>
             <span class="texatom" id="MathJax-Span-376">
              <span class="mrow" id="MathJax-Span-377">
               <span class="mo" id="MathJax-Span-378">
                /
               </span>
              </span>
             </span>
             <span class="mn" id="MathJax-Span-379">
              2
             </span>
             <span class="mo" id="MathJax-Span-380">
              )
             </span>
             <span class="msubsup" id="MathJax-Span-381">
              <span class="mi" id="MathJax-Span-382">
               y
              </span>
              <span class="mi" id="MathJax-Span-383">
               T
              </span>
             </span>
             <span class="msubsup" id="MathJax-Span-384">
              <span class="mi" id="MathJax-Span-385">
               Q
              </span>
              <span class="texatom" id="MathJax-Span-386">
               <span class="mrow" id="MathJax-Span-387">
                <span class="mo" id="MathJax-Span-388">
                 −
                </span>
                <span class="mn" id="MathJax-Span-389">
                 1
                </span>
               </span>
              </span>
             </span>
             <span class="mi" id="MathJax-Span-390">
              y
             </span>
            </span>
           </span>
          </span>
         </span>
        </li>
       </ul>
       <p>
        这个函数的几何意义可以通过下图解释：
        <br/>
        <a href="http://dataunion.org/wp-content/uploads/2015/01/115.png">
         <img src="http://dataunion.org/wp-content/uploads/2015/01/115.png"/>
        </a>
        <br/>
        随着
        <span class="editor-math-element" data-id="JTI1MjQlMjU1Q2JmJTI1MjB5JTI1MjQ=">
         <span class="MathJax" id="MathJax-Element-31-Frame">
          <span class="math" id="MathJax-Span-391">
           <span class="mrow" id="MathJax-Span-392">
            <span class="mi" id="MathJax-Span-393">
             y
            </span>
           </span>
          </span>
         </span>
        </span>
        的变化，
        <span class="editor-math-element" data-id="JTI1MjQlMjU3QiUyNTVDYmYlMjUyMHklMjU3RCUyNTVFVCUyNTdCJTI1NUNiZiUyNTIweCUyNTdEJTI1MjAlMjUyNA==">
         <span class="MathJax" id="MathJax-Element-32-Frame">
          <span class="math" id="MathJax-Span-394">
           <span class="mrow" id="MathJax-Span-395">
            <span class="msubsup" id="MathJax-Span-396">
             <span class="texatom" id="MathJax-Span-397">
              <span class="mrow" id="MathJax-Span-398">
               <span class="mi" id="MathJax-Span-399">
                y
               </span>
              </span>
             </span>
             <span class="mi" id="MathJax-Span-400">
              T
             </span>
            </span>
            <span class="texatom" id="MathJax-Span-401">
             <span class="mrow" id="MathJax-Span-402">
              <span class="mi" id="MathJax-Span-403">
               x
              </span>
             </span>
            </span>
           </span>
          </span>
         </span>
        </span>
        代表了倾斜程度不同的过原点的超平面（上图中就是一条直线）。
        <span class="editor-math-element" data-id="JTI1MjQlMjU3QiUyNTVDYmYlMjUyMHklMjU3RCUyNTVFVCUyNTdCJTI1NUNiZiUyNTIweCUyNTdEJTI1MjAtJTI1MjBmJTI1MjglMjU3QiUyNTVDYmYlMjUyMHglMjU3RCUyNTI5JTI1MjQ=">
         <span class="MathJax" id="MathJax-Element-33-Frame">
          <span class="math" id="MathJax-Span-404">
           <span class="mrow" id="MathJax-Span-405">
            <span class="msubsup" id="MathJax-Span-406">
             <span class="texatom" id="MathJax-Span-407">
              <span class="mrow" id="MathJax-Span-408">
               <span class="mi" id="MathJax-Span-409">
                y
               </span>
              </span>
             </span>
             <span class="mi" id="MathJax-Span-410">
              T
             </span>
            </span>
            <span class="texatom" id="MathJax-Span-411">
             <span class="mrow" id="MathJax-Span-412">
              <span class="mi" id="MathJax-Span-413">
               x
              </span>
             </span>
            </span>
            <span class="mo" id="MathJax-Span-414">
             −
            </span>
            <span class="mi" id="MathJax-Span-415">
             f
            </span>
            <span class="mo" id="MathJax-Span-416">
             (
            </span>
            <span class="texatom" id="MathJax-Span-417">
             <span class="mrow" id="MathJax-Span-418">
              <span class="mi" id="MathJax-Span-419">
               x
              </span>
             </span>
            </span>
            <span class="mo" id="MathJax-Span-420">
             )
            </span>
           </span>
          </span>
         </span>
        </span>
        就是这条线到函数
        <span class="editor-math-element" data-id="JTI1MjRmJTI1MjglMjU3QiUyNTVDYmYlMjUyMHglMjU3RCUyNTI5JTI1MjQ=">
         <span class="MathJax" id="MathJax-Element-34-Frame">
          <span class="math" id="MathJax-Span-421">
           <span class="mrow" id="MathJax-Span-422">
            <span class="mi" id="MathJax-Span-423">
             f
            </span>
            <span class="mo" id="MathJax-Span-424">
             (
            </span>
            <span class="texatom" id="MathJax-Span-425">
             <span class="mrow" id="MathJax-Span-426">
              <span class="mi" id="MathJax-Span-427">
               x
              </span>
             </span>
            </span>
            <span class="mo" id="MathJax-Span-428">
             )
            </span>
           </span>
          </span>
         </span>
        </span>
        的距离。这个距离显然在图中的黑点处取到最大值，也就是最小上界。共轭函数就是描述这个最小上界随着
        <span class="editor-math-element" data-id="JTI1MjQlMjU1Q2JmJTI1MjB5JTI1MjQ=">
         <span class="MathJax" id="MathJax-Element-35-Frame">
          <span class="math" id="MathJax-Span-429">
           <span class="mrow" id="MathJax-Span-430">
            <span class="mi" id="MathJax-Span-431">
             y
            </span>
           </span>
          </span>
         </span>
        </span>
        的变化所变化的情况。
       </p>
       <p>
        很容易看出不管原函数的凹凸性如何，共轭函数一定是凸函数（可以由凸函数性质看出，这里不细说）。
       </p>
       <h2>
        拉格朗日函数
        <span id="anchor_JTI1dTYyQzklMjV1NjgzQyUyNXU2NzE3JTI1dTY1RTUlMjV1NTFGRCUyNXU2NTcwMg==">
        </span>
       </h2>
       <p>
        对于一个标准形式的优化问题：
       </p>
       <div class="MathJax_Display">
        <span class="MathJax" id="MathJax-Element-36-Frame">
         <span class="math" id="MathJax-Span-432">
          <span class="mrow" id="MathJax-Span-433">
           <span class="mtable" id="MathJax-Span-434">
            <span class="mtd" id="MathJax-Span-435">
             <span class="mrow" id="MathJax-Span-436">
              <span class="mtext" id="MathJax-Span-437">
               minimize
              </span>
              <span class="mspace" id="MathJax-Span-438">
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-449">
             <span class="mrow" id="MathJax-Span-450">
              <span class="mtext" id="MathJax-Span-451">
               subject to
              </span>
              <span class="mspace" id="MathJax-Span-452">
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-474">
             <span class="mrow" id="MathJax-Span-475">
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-439">
             <span class="mrow" id="MathJax-Span-440">
              <span class="msubsup" id="MathJax-Span-441">
               <span class="mi" id="MathJax-Span-442">
                f
               </span>
               <span class="mn" id="MathJax-Span-443">
                0
               </span>
              </span>
              <span class="mo" id="MathJax-Span-444">
               (
              </span>
              <span class="texatom" id="MathJax-Span-445">
               <span class="mrow" id="MathJax-Span-446">
                <span class="mi" id="MathJax-Span-447">
                 x
                </span>
               </span>
              </span>
              <span class="mo" id="MathJax-Span-448">
               )
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-453">
             <span class="mrow" id="MathJax-Span-454">
              <span class="msubsup" id="MathJax-Span-455">
               <span class="mi" id="MathJax-Span-456">
                f
               </span>
               <span class="mi" id="MathJax-Span-457">
                i
               </span>
              </span>
              <span class="mo" id="MathJax-Span-458">
               (
              </span>
              <span class="texatom" id="MathJax-Span-459">
               <span class="mrow" id="MathJax-Span-460">
                <span class="mi" id="MathJax-Span-461">
                 x
                </span>
               </span>
              </span>
              <span class="mo" id="MathJax-Span-462">
               )
              </span>
              <span class="mo" id="MathJax-Span-463">
               ≤
              </span>
              <span class="mn" id="MathJax-Span-464">
               0
              </span>
              <span class="mo" id="MathJax-Span-465">
               ,
              </span>
              <span class="mspace" id="MathJax-Span-466">
              </span>
              <span class="mi" id="MathJax-Span-467">
               i
              </span>
              <span class="mo" id="MathJax-Span-468">
               =
              </span>
              <span class="mn" id="MathJax-Span-469">
               1
              </span>
              <span class="mo" id="MathJax-Span-470">
               ,
              </span>
              <span class="mo" id="MathJax-Span-471">
               …
              </span>
              <span class="mo" id="MathJax-Span-472">
               ,
              </span>
              <span class="mi" id="MathJax-Span-473">
               m
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-476">
             <span class="mrow" id="MathJax-Span-477">
              <span class="msubsup" id="MathJax-Span-478">
               <span class="mi" id="MathJax-Span-479">
                h
               </span>
               <span class="mi" id="MathJax-Span-480">
                i
               </span>
              </span>
              <span class="mo" id="MathJax-Span-481">
               (
              </span>
              <span class="texatom" id="MathJax-Span-482">
               <span class="mrow" id="MathJax-Span-483">
                <span class="mi" id="MathJax-Span-484">
                 x
                </span>
               </span>
              </span>
              <span class="mo" id="MathJax-Span-485">
               )
              </span>
              <span class="mo" id="MathJax-Span-486">
               =
              </span>
              <span class="mn" id="MathJax-Span-487">
               0
              </span>
              <span class="mo" id="MathJax-Span-488">
               ,
              </span>
              <span class="mspace" id="MathJax-Span-489">
              </span>
              <span class="mi" id="MathJax-Span-490">
               i
              </span>
              <span class="mo" id="MathJax-Span-491">
               =
              </span>
              <span class="mn" id="MathJax-Span-492">
               1
              </span>
              <span class="mo" id="MathJax-Span-493">
               ,
              </span>
              <span class="mo" id="MathJax-Span-494">
               …
              </span>
              <span class="mo" id="MathJax-Span-495">
               ,
              </span>
              <span class="mi" id="MathJax-Span-496">
               p
              </span>
             </span>
            </span>
           </span>
           <span class="mtext" id="MathJax-Span-497">
           </span>
          </span>
         </span>
        </span>
       </div>
       <p>
        这里我们并没有假定这个问题是凸的。
       </p>
       <p>
        拉格朗日函数就是将目标函数和约束进行有权重的求和：
       </p>
       <div class="MathJax_Display">
        <span class="MathJax" id="MathJax-Element-37-Frame">
         <span class="math" id="MathJax-Span-498">
          <span class="mrow" id="MathJax-Span-499">
           <span class="mi" id="MathJax-Span-500">
            L
           </span>
           <span class="mo" id="MathJax-Span-501">
            (
           </span>
           <span class="texatom" id="MathJax-Span-502">
            <span class="mrow" id="MathJax-Span-503">
             <span class="mi" id="MathJax-Span-504">
              x
             </span>
            </span>
           </span>
           <span class="mo" id="MathJax-Span-505">
            ,
           </span>
           <span class="texatom" id="MathJax-Span-506">
            <span class="mrow" id="MathJax-Span-507">
             <span class="mi" id="MathJax-Span-508">
              λ
             </span>
            </span>
           </span>
           <span class="mo" id="MathJax-Span-509">
            ,
           </span>
           <span class="texatom" id="MathJax-Span-510">
            <span class="mrow" id="MathJax-Span-511">
             <span class="mi" id="MathJax-Span-512">
              ν
             </span>
            </span>
           </span>
           <span class="mo" id="MathJax-Span-513">
            )
           </span>
           <span class="mo" id="MathJax-Span-514">
            =
           </span>
           <span class="msubsup" id="MathJax-Span-515">
            <span class="mi" id="MathJax-Span-516">
             f
            </span>
            <span class="mn" id="MathJax-Span-517">
             0
            </span>
           </span>
           <span class="mo" id="MathJax-Span-518">
            (
           </span>
           <span class="texatom" id="MathJax-Span-519">
            <span class="mrow" id="MathJax-Span-520">
             <span class="mi" id="MathJax-Span-521">
              x
             </span>
            </span>
           </span>
           <span class="mo" id="MathJax-Span-522">
            )
           </span>
           <span class="mo" id="MathJax-Span-523">
            +
           </span>
           <span class="munderover" id="MathJax-Span-524">
            <span class="mo" id="MathJax-Span-525">
             ∑
            </span>
            <span class="texatom" id="MathJax-Span-526">
             <span class="mrow" id="MathJax-Span-527">
              <span class="mi" id="MathJax-Span-528">
               i
              </span>
              <span class="mo" id="MathJax-Span-529">
               =
              </span>
              <span class="mn" id="MathJax-Span-530">
               1
              </span>
             </span>
            </span>
            <span class="mi" id="MathJax-Span-531">
             m
            </span>
           </span>
           <span class="msubsup" id="MathJax-Span-532">
            <span class="mi" id="MathJax-Span-533">
             λ
            </span>
            <span class="mi" id="MathJax-Span-534">
             i
            </span>
           </span>
           <span class="msubsup" id="MathJax-Span-535">
            <span class="mi" id="MathJax-Span-536">
             f
            </span>
            <span class="mi" id="MathJax-Span-537">
             i
            </span>
           </span>
           <span class="mo" id="MathJax-Span-538">
            (
           </span>
           <span class="texatom" id="MathJax-Span-539">
            <span class="mrow" id="MathJax-Span-540">
             <span class="mi" id="MathJax-Span-541">
              x
             </span>
            </span>
           </span>
           <span class="mo" id="MathJax-Span-542">
            )
           </span>
           <span class="mo" id="MathJax-Span-543">
            +
           </span>
           <span class="munderover" id="MathJax-Span-544">
            <span class="mo" id="MathJax-Span-545">
             ∑
            </span>
            <span class="texatom" id="MathJax-Span-546">
             <span class="mrow" id="MathJax-Span-547">
              <span class="mi" id="MathJax-Span-548">
               i
              </span>
              <span class="mo" id="MathJax-Span-549">
               =
              </span>
              <span class="mn" id="MathJax-Span-550">
               1
              </span>
             </span>
            </span>
            <span class="mi" id="MathJax-Span-551">
             p
            </span>
           </span>
           <span class="msubsup" id="MathJax-Span-552">
            <span class="mi" id="MathJax-Span-553">
             ν
            </span>
            <span class="mi" id="MathJax-Span-554">
             i
            </span>
           </span>
           <span class="msubsup" id="MathJax-Span-555">
            <span class="mi" id="MathJax-Span-556">
             h
            </span>
            <span class="mi" id="MathJax-Span-557">
             i
            </span>
           </span>
           <span class="mo" id="MathJax-Span-558">
            (
           </span>
           <span class="texatom" id="MathJax-Span-559">
            <span class="mrow" id="MathJax-Span-560">
             <span class="mi" id="MathJax-Span-561">
              x
             </span>
            </span>
           </span>
           <span class="mo" id="MathJax-Span-562">
            )
           </span>
           <span class="mtext" id="MathJax-Span-563">
           </span>
          </span>
         </span>
        </span>
       </div>
       <p>
        这个函数不仅仅是
        <span class="editor-math-element" data-id="JTI1MjQlMjU1Q2JmJTI1MjB4JTI1MjQ=">
         <span class="MathJax" id="MathJax-Element-38-Frame">
          <span class="math" id="MathJax-Span-564">
           <span class="mrow" id="MathJax-Span-565">
            <span class="mi" id="MathJax-Span-566">
             x
            </span>
           </span>
          </span>
         </span>
        </span>
        的函数，还是拉格朗日乘子向量
        <span class="editor-math-element" data-id="JTI1MjQlMjU1Q2JmJTI1MjAlMjU1Q2xhbWJkYSUyNTI0">
         <span class="MathJax" id="MathJax-Element-39-Frame">
          <span class="math" id="MathJax-Span-567">
           <span class="mrow" id="MathJax-Span-568">
            <span class="mi" id="MathJax-Span-569">
             λ
            </span>
           </span>
          </span>
         </span>
        </span>
        和
        <span class="editor-math-element" data-id="JTI1MjQlMjU1Q2JmJTI1MjAlMjU1Q251JTI1MjQ=">
         <span class="MathJax" id="MathJax-Element-40-Frame">
          <span class="math" id="MathJax-Span-570">
           <span class="mrow" id="MathJax-Span-571">
            <span class="mi" id="MathJax-Span-572">
             ν
            </span>
           </span>
          </span>
         </span>
        </span>
        的函数。其中
        <span class="editor-math-element" data-id="JTI1MjQlMjU1Q2xhbWJkYV9pJTI1MjQ=">
         <span class="MathJax" id="MathJax-Element-41-Frame">
          <span class="math" id="MathJax-Span-573">
           <span class="mrow" id="MathJax-Span-574">
            <span class="msubsup" id="MathJax-Span-575">
             <span class="mi" id="MathJax-Span-576">
              λ
             </span>
             <span class="mi" id="MathJax-Span-577">
              i
             </span>
            </span>
           </span>
          </span>
         </span>
        </span>
        被称为对应于不等约束
        <span class="editor-math-element" data-id="JTI1MjRmX2klMjUyOCUyNTdCJTI1NUNiZiUyNTIweCUyNTdEJTI1MjklMjUyMCUyNTVDbGUlMjUyMDAlMjUyNA==">
         <span class="MathJax" id="MathJax-Element-42-Frame">
          <span class="math" id="MathJax-Span-578">
           <span class="mrow" id="MathJax-Span-579">
            <span class="msubsup" id="MathJax-Span-580">
             <span class="mi" id="MathJax-Span-581">
              f
             </span>
             <span class="mi" id="MathJax-Span-582">
              i
             </span>
            </span>
            <span class="mo" id="MathJax-Span-583">
             (
            </span>
            <span class="texatom" id="MathJax-Span-584">
             <span class="mrow" id="MathJax-Span-585">
              <span class="mi" id="MathJax-Span-586">
               x
              </span>
             </span>
            </span>
            <span class="mo" id="MathJax-Span-587">
             )
            </span>
            <span class="mo" id="MathJax-Span-588">
             ≤
            </span>
            <span class="mn" id="MathJax-Span-589">
             0
            </span>
           </span>
          </span>
         </span>
        </span>
        的拉格朗日乘子；
        <span class="editor-math-element" data-id="JTI1MjQlMjU1Q251X2klMjUyNA==">
         <span class="MathJax" id="MathJax-Element-43-Frame">
          <span class="math" id="MathJax-Span-590">
           <span class="mrow" id="MathJax-Span-591">
            <span class="msubsup" id="MathJax-Span-592">
             <span class="mi" id="MathJax-Span-593">
              ν
             </span>
             <span class="mi" id="MathJax-Span-594">
              i
             </span>
            </span>
           </span>
          </span>
         </span>
        </span>
        被称为对应于相等约束
        <span class="editor-math-element" data-id="JTI1MjRoX2klMjUyOCUyNTdCJTI1NUNiZiUyNTIweCUyNTdEJTI1MjklMjUyMCUyNTNEJTI1MjAwJTI1MjQ=">
         <span class="MathJax" id="MathJax-Element-44-Frame">
          <span class="math" id="MathJax-Span-595">
           <span class="mrow" id="MathJax-Span-596">
            <span class="msubsup" id="MathJax-Span-597">
             <span class="mi" id="MathJax-Span-598">
              h
             </span>
             <span class="mi" id="MathJax-Span-599">
              i
             </span>
            </span>
            <span class="mo" id="MathJax-Span-600">
             (
            </span>
            <span class="texatom" id="MathJax-Span-601">
             <span class="mrow" id="MathJax-Span-602">
              <span class="mi" id="MathJax-Span-603">
               x
              </span>
             </span>
            </span>
            <span class="mo" id="MathJax-Span-604">
             )
            </span>
            <span class="mo" id="MathJax-Span-605">
             =
            </span>
            <span class="mn" id="MathJax-Span-606">
             0
            </span>
           </span>
          </span>
         </span>
        </span>
        的拉格朗日乘子。
       </p>
       <h2>
        拉格朗日对偶函数
        <span id="anchor_JTI1dTYyQzklMjV1NjgzQyUyNXU2NzE3JTI1dTY1RTUlMjV1NUJGOSUyNXU1MDc2JTI1dTUxRkQlMjV1NjU3MDM=">
        </span>
       </h2>
       <p>
        拉格朗日对偶函数，或者直接叫对偶函数，被定义为拉格朗日函数在
        <span class="editor-math-element" data-id="JTI1MjQlMjU1Q2JmJTI1MjB4JTI1MjQ=">
         <span class="MathJax" id="MathJax-Element-45-Frame">
          <span class="math" id="MathJax-Span-607">
           <span class="mrow" id="MathJax-Span-608">
            <span class="mi" id="MathJax-Span-609">
             x
            </span>
           </span>
          </span>
         </span>
        </span>
        自由变化时所取到的最小值：
       </p>
       <div class="MathJax_Display">
        <span class="MathJax" id="MathJax-Element-46-Frame">
         <span class="math" id="MathJax-Span-610">
          <span class="mrow" id="MathJax-Span-611">
           <span class="mi" id="MathJax-Span-612">
            g
           </span>
           <span class="mo" id="MathJax-Span-613">
            (
           </span>
           <span class="texatom" id="MathJax-Span-614">
            <span class="mrow" id="MathJax-Span-615">
             <span class="mi" id="MathJax-Span-616">
              λ
             </span>
            </span>
           </span>
           <span class="mo" id="MathJax-Span-617">
            ,
           </span>
           <span class="texatom" id="MathJax-Span-618">
            <span class="mrow" id="MathJax-Span-619">
             <span class="mi" id="MathJax-Span-620">
              ν
             </span>
            </span>
           </span>
           <span class="mo" id="MathJax-Span-621">
            )
           </span>
           <span class="mo" id="MathJax-Span-622">
            =
           </span>
           <span class="munderover" id="MathJax-Span-623">
            <span class="mo" id="MathJax-Span-624">
             inf
            </span>
            <span class="texatom" id="MathJax-Span-625">
             <span class="mrow" id="MathJax-Span-626">
              <span class="texatom" id="MathJax-Span-627">
               <span class="mrow" id="MathJax-Span-628">
                <span class="mi" id="MathJax-Span-629">
                 x
                </span>
               </span>
              </span>
              <span class="mo" id="MathJax-Span-630">
               ∈
              </span>
              <span class="texatom" id="MathJax-Span-631">
               <span class="mrow" id="MathJax-Span-632">
                <span class="mi" id="MathJax-Span-633">
                 D
                </span>
               </span>
              </span>
             </span>
            </span>
           </span>
           <span class="mo" id="MathJax-Span-634">
            (
           </span>
           <span class="msubsup" id="MathJax-Span-635">
            <span class="mi" id="MathJax-Span-636">
             f
            </span>
            <span class="mn" id="MathJax-Span-637">
             0
            </span>
           </span>
           <span class="mo" id="MathJax-Span-638">
            (
           </span>
           <span class="texatom" id="MathJax-Span-639">
            <span class="mrow" id="MathJax-Span-640">
             <span class="mi" id="MathJax-Span-641">
              x
             </span>
            </span>
           </span>
           <span class="mo" id="MathJax-Span-642">
            )
           </span>
           <span class="mo" id="MathJax-Span-643">
            +
           </span>
           <span class="munderover" id="MathJax-Span-644">
            <span class="mo" id="MathJax-Span-645">
             ∑
            </span>
            <span class="texatom" id="MathJax-Span-646">
             <span class="mrow" id="MathJax-Span-647">
              <span class="mi" id="MathJax-Span-648">
               i
              </span>
              <span class="mo" id="MathJax-Span-649">
               =
              </span>
              <span class="mn" id="MathJax-Span-650">
               1
              </span>
             </span>
            </span>
            <span class="mi" id="MathJax-Span-651">
             m
            </span>
           </span>
           <span class="msubsup" id="MathJax-Span-652">
            <span class="mi" id="MathJax-Span-653">
             λ
            </span>
            <span class="mi" id="MathJax-Span-654">
             i
            </span>
           </span>
           <span class="msubsup" id="MathJax-Span-655">
            <span class="mi" id="MathJax-Span-656">
             f
            </span>
            <span class="mi" id="MathJax-Span-657">
             i
            </span>
           </span>
           <span class="mo" id="MathJax-Span-658">
            (
           </span>
           <span class="texatom" id="MathJax-Span-659">
            <span class="mrow" id="MathJax-Span-660">
             <span class="mi" id="MathJax-Span-661">
              x
             </span>
            </span>
           </span>
           <span class="mo" id="MathJax-Span-662">
            )
           </span>
           <span class="mo" id="MathJax-Span-663">
            +
           </span>
           <span class="munderover" id="MathJax-Span-664">
            <span class="mo" id="MathJax-Span-665">
             ∑
            </span>
            <span class="texatom" id="MathJax-Span-666">
             <span class="mrow" id="MathJax-Span-667">
              <span class="mi" id="MathJax-Span-668">
               i
              </span>
              <span class="mo" id="MathJax-Span-669">
               =
              </span>
              <span class="mn" id="MathJax-Span-670">
               1
              </span>
             </span>
            </span>
            <span class="mi" id="MathJax-Span-671">
             p
            </span>
           </span>
           <span class="msubsup" id="MathJax-Span-672">
            <span class="mi" id="MathJax-Span-673">
             ν
            </span>
            <span class="mi" id="MathJax-Span-674">
             i
            </span>
           </span>
           <span class="msubsup" id="MathJax-Span-675">
            <span class="mi" id="MathJax-Span-676">
             h
            </span>
            <span class="mi" id="MathJax-Span-677">
             i
            </span>
           </span>
           <span class="mo" id="MathJax-Span-678">
            (
           </span>
           <span class="texatom" id="MathJax-Span-679">
            <span class="mrow" id="MathJax-Span-680">
             <span class="mi" id="MathJax-Span-681">
              x
             </span>
            </span>
           </span>
           <span class="mo" id="MathJax-Span-682">
            )
           </span>
           <span class="mo" id="MathJax-Span-683">
            )
           </span>
           <span class="mtext" id="MathJax-Span-684">
           </span>
          </span>
         </span>
        </span>
       </div>
       <p>
        很显然，这是一个关于拉格朗日乘子向量
        <span class="editor-math-element" data-id="JTI1MjQlMjU1Q2JmJTI1MjAlMjU1Q2xhbWJkYSUyNTJDJTI1MjAlMjU1Q251JTI1MjQ=">
         <span class="MathJax" id="MathJax-Element-47-Frame">
          <span class="math" id="MathJax-Span-685">
           <span class="mrow" id="MathJax-Span-686">
            <span class="mi" id="MathJax-Span-687">
             λ
            </span>
            <span class="mo" id="MathJax-Span-688">
             ,
            </span>
            <span class="mi" id="MathJax-Span-689">
             ν
            </span>
           </span>
          </span>
         </span>
        </span>
        的函数。可以通过凸函数的性质发现无论原优化问题的凹凸性，这个对偶函数始终是凹的。
       </p>
       <h2>
        目标函数最优值的下界
        <span id="anchor_JTI1dTc2RUUlMjV1NjgwNyUyNXU1MUZEJTI1dTY1NzAlMjV1NjcwMCUyNXU0RjE4JTI1dTUwM0MlMjV1NzY4NCUyNXU0RTBCJTI1dTc1NEM0">
        </span>
       </h2>
       <p>
        假设原始问题目标函数最优值是
        <span class="editor-math-element" data-id="JTI1MjRwJTI1NUUlMjU1Q3N0YXIlMjUyNA==">
         <span class="MathJax" id="MathJax-Element-48-Frame">
          <span class="math" id="MathJax-Span-690">
           <span class="mrow" id="MathJax-Span-691">
            <span class="msubsup" id="MathJax-Span-692">
             <span class="mi" id="MathJax-Span-693">
              p
             </span>
             <span class="mo" id="MathJax-Span-694">
              ⋆
             </span>
            </span>
           </span>
          </span>
         </span>
        </span>
        。如果令
        <span class="editor-math-element" data-id="JTI1MjQlMjU1Q2xhbWJkYSUyNTIwJTI1NUNnZSUyNTIwMCUyNTI0">
         <span class="MathJax" id="MathJax-Element-49-Frame">
          <span class="math" id="MathJax-Span-695">
           <span class="mrow" id="MathJax-Span-696">
            <span class="mi" id="MathJax-Span-697">
             λ
            </span>
            <span class="mo" id="MathJax-Span-698">
             ≥
            </span>
            <span class="mn" id="MathJax-Span-699">
             0
            </span>
           </span>
          </span>
         </span>
        </span>
        ，则对于任何一个原问题的可行解
        <span class="editor-math-element" data-id="JTI1MjQlMjU1Q3RpbGRlJTI1MjAlMjU3QiUyNTVDYmYlMjUyMHglMjU3RCUyNTI0">
         <span class="MathJax" id="MathJax-Element-50-Frame">
          <span class="math" id="MathJax-Span-700">
           <span class="mrow" id="MathJax-Span-701">
            <span class="texatom" id="MathJax-Span-702">
             <span class="mrow" id="MathJax-Span-703">
              <span class="munderover" id="MathJax-Span-704">
               <span class="mi" id="MathJax-Span-705">
                x
               </span>
               <span class="mo" id="MathJax-Span-706">
                ~
               </span>
              </span>
             </span>
            </span>
           </span>
          </span>
         </span>
        </span>
        ，都有
        <span class="editor-math-element" data-id="JTI1MjRmX2klMjUyOCUyNTVDdGlsZGUlMjUyMCUyNTdCJTI1NUNiZiUyNTIweCUyNTdEJTI1MjklMjUyMCUyNTVDbGUlMjUyMDAlMjUyQyUyNTIwaF9pJTI1MjglMjU1Q3RpbGRlJTI1MjAlMjU3QiUyNTVDYmYlMjUyMHglMjU3RCUyNTI5JTI1MjAlMjUzRCUyNTIwMCUyNTI0">
         <span class="MathJax" id="MathJax-Element-51-Frame">
          <span class="math" id="MathJax-Span-707">
           <span class="mrow" id="MathJax-Span-708">
            <span class="msubsup" id="MathJax-Span-709">
             <span class="mi" id="MathJax-Span-710">
              f
             </span>
             <span class="mi" id="MathJax-Span-711">
              i
             </span>
            </span>
            <span class="mo" id="MathJax-Span-712">
             (
            </span>
            <span class="texatom" id="MathJax-Span-713">
             <span class="mrow" id="MathJax-Span-714">
              <span class="munderover" id="MathJax-Span-715">
               <span class="mi" id="MathJax-Span-716">
                x
               </span>
               <span class="mo" id="MathJax-Span-717">
                ~
               </span>
              </span>
             </span>
            </span>
            <span class="mo" id="MathJax-Span-718">
             )
            </span>
            <span class="mo" id="MathJax-Span-719">
             ≤
            </span>
            <span class="mn" id="MathJax-Span-720">
             0
            </span>
            <span class="mo" id="MathJax-Span-721">
             ,
            </span>
            <span class="msubsup" id="MathJax-Span-722">
             <span class="mi" id="MathJax-Span-723">
              h
             </span>
             <span class="mi" id="MathJax-Span-724">
              i
             </span>
            </span>
            <span class="mo" id="MathJax-Span-725">
             (
            </span>
            <span class="texatom" id="MathJax-Span-726">
             <span class="mrow" id="MathJax-Span-727">
              <span class="munderover" id="MathJax-Span-728">
               <span class="mi" id="MathJax-Span-729">
                x
               </span>
               <span class="mo" id="MathJax-Span-730">
                ~
               </span>
              </span>
             </span>
            </span>
            <span class="mo" id="MathJax-Span-731">
             )
            </span>
            <span class="mo" id="MathJax-Span-732">
             =
            </span>
            <span class="mn" id="MathJax-Span-733">
             0
            </span>
           </span>
          </span>
         </span>
        </span>
        。我们很容易发现：
       </p>
       <div class="MathJax_Display">
        <span class="MathJax" id="MathJax-Element-52-Frame">
         <span class="math" id="MathJax-Span-734">
          <span class="mrow" id="MathJax-Span-735">
           <span class="munderover" id="MathJax-Span-736">
            <span class="mo" id="MathJax-Span-737">
             ∑
            </span>
            <span class="texatom" id="MathJax-Span-738">
             <span class="mrow" id="MathJax-Span-739">
              <span class="mi" id="MathJax-Span-740">
               i
              </span>
              <span class="mo" id="MathJax-Span-741">
               =
              </span>
              <span class="mn" id="MathJax-Span-742">
               1
              </span>
             </span>
            </span>
            <span class="mi" id="MathJax-Span-743">
             m
            </span>
           </span>
           <span class="msubsup" id="MathJax-Span-744">
            <span class="mi" id="MathJax-Span-745">
             λ
            </span>
            <span class="mi" id="MathJax-Span-746">
             i
            </span>
           </span>
           <span class="msubsup" id="MathJax-Span-747">
            <span class="mi" id="MathJax-Span-748">
             f
            </span>
            <span class="mi" id="MathJax-Span-749">
             i
            </span>
           </span>
           <span class="mo" id="MathJax-Span-750">
            (
           </span>
           <span class="texatom" id="MathJax-Span-751">
            <span class="mrow" id="MathJax-Span-752">
             <span class="munderover" id="MathJax-Span-753">
              <span class="mi" id="MathJax-Span-754">
               x
              </span>
              <span class="mo" id="MathJax-Span-755">
               ~
              </span>
             </span>
            </span>
           </span>
           <span class="mo" id="MathJax-Span-756">
            )
           </span>
           <span class="mo" id="MathJax-Span-757">
            +
           </span>
           <span class="munderover" id="MathJax-Span-758">
            <span class="mo" id="MathJax-Span-759">
             ∑
            </span>
            <span class="texatom" id="MathJax-Span-760">
             <span class="mrow" id="MathJax-Span-761">
              <span class="mi" id="MathJax-Span-762">
               i
              </span>
              <span class="mo" id="MathJax-Span-763">
               =
              </span>
              <span class="mn" id="MathJax-Span-764">
               1
              </span>
             </span>
            </span>
            <span class="mi" id="MathJax-Span-765">
             p
            </span>
           </span>
           <span class="msubsup" id="MathJax-Span-766">
            <span class="mi" id="MathJax-Span-767">
             ν
            </span>
            <span class="mi" id="MathJax-Span-768">
             i
            </span>
           </span>
           <span class="msubsup" id="MathJax-Span-769">
            <span class="mi" id="MathJax-Span-770">
             h
            </span>
            <span class="mi" id="MathJax-Span-771">
             i
            </span>
           </span>
           <span class="mo" id="MathJax-Span-772">
            (
           </span>
           <span class="texatom" id="MathJax-Span-773">
            <span class="mrow" id="MathJax-Span-774">
             <span class="munderover" id="MathJax-Span-775">
              <span class="mi" id="MathJax-Span-776">
               x
              </span>
              <span class="mo" id="MathJax-Span-777">
               ~
              </span>
             </span>
            </span>
           </span>
           <span class="mo" id="MathJax-Span-778">
            )
           </span>
           <span class="mo" id="MathJax-Span-779">
            ≤
           </span>
           <span class="mn" id="MathJax-Span-780">
            0
           </span>
           <span class="mtext" id="MathJax-Span-781">
           </span>
          </span>
         </span>
        </span>
       </div>
       <p>
        因此，很容易得到下面的不等式：
       </p>
       <div class="MathJax_Display">
        <span class="MathJax" id="MathJax-Element-53-Frame">
         <span class="math" id="MathJax-Span-782">
          <span class="mrow" id="MathJax-Span-783">
           <span class="mi" id="MathJax-Span-784">
            g
           </span>
           <span class="mo" id="MathJax-Span-785">
            (
           </span>
           <span class="texatom" id="MathJax-Span-786">
            <span class="mrow" id="MathJax-Span-787">
             <span class="mi" id="MathJax-Span-788">
              λ
             </span>
            </span>
           </span>
           <span class="mo" id="MathJax-Span-789">
            ,
           </span>
           <span class="texatom" id="MathJax-Span-790">
            <span class="mrow" id="MathJax-Span-791">
             <span class="mi" id="MathJax-Span-792">
              ν
             </span>
            </span>
           </span>
           <span class="mo" id="MathJax-Span-793">
            )
           </span>
           <span class="mo" id="MathJax-Span-794">
            =
           </span>
           <span class="munderover" id="MathJax-Span-795">
            <span class="mo" id="MathJax-Span-796">
             inf
            </span>
            <span class="texatom" id="MathJax-Span-797">
             <span class="mrow" id="MathJax-Span-798">
              <span class="texatom" id="MathJax-Span-799">
               <span class="mrow" id="MathJax-Span-800">
                <span class="mi" id="MathJax-Span-801">
                 x
                </span>
               </span>
              </span>
              <span class="mo" id="MathJax-Span-802">
               ∈
              </span>
              <span class="texatom" id="MathJax-Span-803">
               <span class="mrow" id="MathJax-Span-804">
                <span class="mi" id="MathJax-Span-805">
                 D
                </span>
               </span>
              </span>
             </span>
            </span>
           </span>
           <span class="mi" id="MathJax-Span-806">
            L
           </span>
           <span class="mo" id="MathJax-Span-807">
            (
           </span>
           <span class="texatom" id="MathJax-Span-808">
            <span class="mrow" id="MathJax-Span-809">
             <span class="mi" id="MathJax-Span-810">
              x
             </span>
            </span>
           </span>
           <span class="mo" id="MathJax-Span-811">
            ,
           </span>
           <span class="texatom" id="MathJax-Span-812">
            <span class="mrow" id="MathJax-Span-813">
             <span class="mi" id="MathJax-Span-814">
              λ
             </span>
            </span>
           </span>
           <span class="mo" id="MathJax-Span-815">
            ,
           </span>
           <span class="texatom" id="MathJax-Span-816">
            <span class="mrow" id="MathJax-Span-817">
             <span class="mi" id="MathJax-Span-818">
              ν
             </span>
            </span>
           </span>
           <span class="mo" id="MathJax-Span-819">
            )
           </span>
           <span class="mo" id="MathJax-Span-820">
            ≤
           </span>
           <span class="mi" id="MathJax-Span-821">
            L
           </span>
           <span class="mo" id="MathJax-Span-822">
            (
           </span>
           <span class="texatom" id="MathJax-Span-823">
            <span class="mrow" id="MathJax-Span-824">
             <span class="munderover" id="MathJax-Span-825">
              <span class="mi" id="MathJax-Span-826">
               x
              </span>
              <span class="mo" id="MathJax-Span-827">
               ~
              </span>
             </span>
            </span>
           </span>
           <span class="mo" id="MathJax-Span-828">
            ,
           </span>
           <span class="texatom" id="MathJax-Span-829">
            <span class="mrow" id="MathJax-Span-830">
             <span class="mi" id="MathJax-Span-831">
              λ
             </span>
            </span>
           </span>
           <span class="mo" id="MathJax-Span-832">
            ,
           </span>
           <span class="texatom" id="MathJax-Span-833">
            <span class="mrow" id="MathJax-Span-834">
             <span class="mi" id="MathJax-Span-835">
              ν
             </span>
            </span>
           </span>
           <span class="mo" id="MathJax-Span-836">
            )
           </span>
           <span class="mo" id="MathJax-Span-837">
            ≤
           </span>
           <span class="msubsup" id="MathJax-Span-838">
            <span class="mi" id="MathJax-Span-839">
             f
            </span>
            <span class="mn" id="MathJax-Span-840">
             0
            </span>
           </span>
           <span class="mo" id="MathJax-Span-841">
            (
           </span>
           <span class="texatom" id="MathJax-Span-842">
            <span class="mrow" id="MathJax-Span-843">
             <span class="munderover" id="MathJax-Span-844">
              <span class="mi" id="MathJax-Span-845">
               x
              </span>
              <span class="mo" id="MathJax-Span-846">
               ~
              </span>
             </span>
            </span>
           </span>
           <span class="mo" id="MathJax-Span-847">
            )
           </span>
           <span class="mtext" id="MathJax-Span-848">
           </span>
          </span>
         </span>
        </span>
       </div>
       <p>
        所以，我们可以得到：
       </p>
       <div class="MathJax_Display">
        <span class="MathJax" id="MathJax-Element-54-Frame">
         <span class="math" id="MathJax-Span-849">
          <span class="mrow" id="MathJax-Span-850">
           <span class="mi" id="MathJax-Span-851">
            g
           </span>
           <span class="mo" id="MathJax-Span-852">
            (
           </span>
           <span class="texatom" id="MathJax-Span-853">
            <span class="mrow" id="MathJax-Span-854">
             <span class="mi" id="MathJax-Span-855">
              λ
             </span>
            </span>
           </span>
           <span class="mo" id="MathJax-Span-856">
            ,
           </span>
           <span class="texatom" id="MathJax-Span-857">
            <span class="mrow" id="MathJax-Span-858">
             <span class="mi" id="MathJax-Span-859">
              ν
             </span>
            </span>
           </span>
           <span class="mo" id="MathJax-Span-860">
            )
           </span>
           <span class="mo" id="MathJax-Span-861">
            ≤
           </span>
           <span class="msubsup" id="MathJax-Span-862">
            <span class="mi" id="MathJax-Span-863">
             p
            </span>
            <span class="mo" id="MathJax-Span-864">
             ⋆
            </span>
           </span>
           <span class="mtext" id="MathJax-Span-865">
           </span>
          </span>
         </span>
        </span>
       </div>
       <p>
        但是需要注意的是，由于
        <span class="editor-math-element" data-id="JTI1MjRnJTI1MjglMjU3QiUyNTVDYmYlMjUyMCUyNTVDbGFtYmRhJTI1N0QlMjUyQyUyNTdCJTI1NUNiZiUyNTIwJTI1NUNudSUyNTdEJTI1MjklMjUyNA==">
         <span class="MathJax" id="MathJax-Element-55-Frame">
          <span class="math" id="MathJax-Span-866">
           <span class="mrow" id="MathJax-Span-867">
            <span class="mi" id="MathJax-Span-868">
             g
            </span>
            <span class="mo" id="MathJax-Span-869">
             (
            </span>
            <span class="texatom" id="MathJax-Span-870">
             <span class="mrow" id="MathJax-Span-871">
              <span class="mi" id="MathJax-Span-872">
               λ
              </span>
             </span>
            </span>
            <span class="mo" id="MathJax-Span-873">
             ,
            </span>
            <span class="texatom" id="MathJax-Span-874">
             <span class="mrow" id="MathJax-Span-875">
              <span class="mi" id="MathJax-Span-876">
               ν
              </span>
             </span>
            </span>
            <span class="mo" id="MathJax-Span-877">
             )
            </span>
           </span>
          </span>
         </span>
        </span>
        可能取到
        <span class="editor-math-element" data-id="JTI1MjQtJTI1NUNpbmZ0eSUyNTI0">
         <span class="MathJax" id="MathJax-Element-56-Frame">
          <span class="math" id="MathJax-Span-878">
           <span class="mrow" id="MathJax-Span-879">
            <span class="mo" id="MathJax-Span-880">
             −
            </span>
            <span class="mi" id="MathJax-Span-881">
             ∞
            </span>
           </span>
          </span>
         </span>
        </span>
        ，此时这个下界没有任何意义。所以我们下面研究的问题都是在
        <span class="editor-math-element" data-id="JTI1MjRnJTI1MjQ=">
         <span class="MathJax" id="MathJax-Element-57-Frame">
          <span class="math" id="MathJax-Span-882">
           <span class="mrow" id="MathJax-Span-883">
            <span class="mi" id="MathJax-Span-884">
             g
            </span>
           </span>
          </span>
         </span>
        </span>
        函数不会取到无穷这样的定义域内进行的。
       </p>
       <h2>
        拉格朗日对偶函数与共轭函数的联系
        <span id="anchor_JTI1dTYyQzklMjV1NjgzQyUyNXU2NzE3JTI1dTY1RTUlMjV1NUJGOSUyNXU1MDc2JTI1dTUxRkQlMjV1NjU3MCUyNXU0RTBFJTI1dTUxNzElMjV1OEY2RCUyNXU1MUZEJTI1dTY1NzAlMjV1NzY4NCUyNXU4MDU0JTI1dTdDRkI1">
        </span>
       </h2>
       <p>
        线性约束的问题的拉格朗日对偶函数可以通过对共轭函数来表达出来。考虑如下线性约束问题：
       </p>
       <div class="MathJax_Display">
        <span class="MathJax" id="MathJax-Element-58-Frame">
         <span class="math" id="MathJax-Span-885">
          <span class="mrow" id="MathJax-Span-886">
           <span class="mtable" id="MathJax-Span-887">
            <span class="mtd" id="MathJax-Span-888">
             <span class="mrow" id="MathJax-Span-889">
              <span class="mtext" id="MathJax-Span-890">
               minimize
              </span>
              <span class="mspace" id="MathJax-Span-891">
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-902">
             <span class="mrow" id="MathJax-Span-903">
              <span class="mtext" id="MathJax-Span-904">
               subject to
              </span>
              <span class="mspace" id="MathJax-Span-905">
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-916">
             <span class="mrow" id="MathJax-Span-917">
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-892">
             <span class="mrow" id="MathJax-Span-893">
              <span class="msubsup" id="MathJax-Span-894">
               <span class="mi" id="MathJax-Span-895">
                f
               </span>
               <span class="mn" id="MathJax-Span-896">
                0
               </span>
              </span>
              <span class="mo" id="MathJax-Span-897">
               (
              </span>
              <span class="texatom" id="MathJax-Span-898">
               <span class="mrow" id="MathJax-Span-899">
                <span class="mi" id="MathJax-Span-900">
                 x
                </span>
               </span>
              </span>
              <span class="mo" id="MathJax-Span-901">
               )
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-906">
             <span class="mrow" id="MathJax-Span-907">
              <span class="texatom" id="MathJax-Span-908">
               <span class="mrow" id="MathJax-Span-909">
                <span class="mi" id="MathJax-Span-910">
                 A
                </span>
                <span class="mi" id="MathJax-Span-911">
                 x
                </span>
               </span>
              </span>
              <span class="mo" id="MathJax-Span-912">
               ≤
              </span>
              <span class="texatom" id="MathJax-Span-913">
               <span class="mrow" id="MathJax-Span-914">
                <span class="mi" id="MathJax-Span-915">
                 b
                </span>
               </span>
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-918">
             <span class="mrow" id="MathJax-Span-919">
              <span class="texatom" id="MathJax-Span-920">
               <span class="mrow" id="MathJax-Span-921">
                <span class="mi" id="MathJax-Span-922">
                 C
                </span>
                <span class="mi" id="MathJax-Span-923">
                 x
                </span>
               </span>
              </span>
              <span class="mo" id="MathJax-Span-924">
               =
              </span>
              <span class="texatom" id="MathJax-Span-925">
               <span class="mrow" id="MathJax-Span-926">
                <span class="mi" id="MathJax-Span-927">
                 d
                </span>
               </span>
              </span>
             </span>
            </span>
           </span>
           <span class="mtext" id="MathJax-Span-928">
           </span>
          </span>
         </span>
        </span>
       </div>
       <p>
        他的拉格朗日对偶函数为：
       </p>
       <div class="MathJax_Display">
        <span class="MathJax" id="MathJax-Element-59-Frame">
         <span class="math" id="MathJax-Span-929">
          <span class="mrow" id="MathJax-Span-930">
           <span class="mtable" id="MathJax-Span-931">
            <span class="mtd" id="MathJax-Span-932">
             <span class="mrow" id="MathJax-Span-933">
              <span class="mi" id="MathJax-Span-934">
               g
              </span>
              <span class="mo" id="MathJax-Span-935">
               (
              </span>
              <span class="texatom" id="MathJax-Span-936">
               <span class="mrow" id="MathJax-Span-937">
                <span class="mi" id="MathJax-Span-938">
                 λ
                </span>
               </span>
              </span>
              <span class="mo" id="MathJax-Span-939">
               ,
              </span>
              <span class="texatom" id="MathJax-Span-940">
               <span class="mrow" id="MathJax-Span-941">
                <span class="mi" id="MathJax-Span-942">
                 ν
                </span>
               </span>
              </span>
              <span class="mo" id="MathJax-Span-943">
               )
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-994">
             <span class="mrow" id="MathJax-Span-995">
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1046">
             <span class="mrow" id="MathJax-Span-1047">
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-944">
             <span class="mrow" id="MathJax-Span-945">
              <span class="mo" id="MathJax-Span-946">
               =
              </span>
              <span class="munderover" id="MathJax-Span-947">
               <span class="mo" id="MathJax-Span-948">
                inf
               </span>
               <span class="texatom" id="MathJax-Span-949">
                <span class="mrow" id="MathJax-Span-950">
                 <span class="mi" id="MathJax-Span-951">
                  x
                 </span>
                </span>
               </span>
              </span>
              <span class="mo" id="MathJax-Span-952">
               (
              </span>
              <span class="msubsup" id="MathJax-Span-953">
               <span class="mi" id="MathJax-Span-954">
                f
               </span>
               <span class="mn" id="MathJax-Span-955">
                0
               </span>
              </span>
              <span class="mo" id="MathJax-Span-956">
               (
              </span>
              <span class="texatom" id="MathJax-Span-957">
               <span class="mrow" id="MathJax-Span-958">
                <span class="mi" id="MathJax-Span-959">
                 x
                </span>
               </span>
              </span>
              <span class="mo" id="MathJax-Span-960">
               )
              </span>
              <span class="mo" id="MathJax-Span-961">
               +
              </span>
              <span class="msubsup" id="MathJax-Span-962">
               <span class="texatom" id="MathJax-Span-963">
                <span class="mrow" id="MathJax-Span-964">
                 <span class="mi" id="MathJax-Span-965">
                  λ
                 </span>
                </span>
               </span>
               <span class="mi" id="MathJax-Span-966">
                T
               </span>
              </span>
              <span class="mo" id="MathJax-Span-967">
               (
              </span>
              <span class="texatom" id="MathJax-Span-968">
               <span class="mrow" id="MathJax-Span-969">
                <span class="mi" id="MathJax-Span-970">
                 A
                </span>
                <span class="mi" id="MathJax-Span-971">
                 x
                </span>
               </span>
              </span>
              <span class="mo" id="MathJax-Span-972">
               −
              </span>
              <span class="texatom" id="MathJax-Span-973">
               <span class="mrow" id="MathJax-Span-974">
                <span class="mi" id="MathJax-Span-975">
                 b
                </span>
               </span>
              </span>
              <span class="mo" id="MathJax-Span-976">
               )
              </span>
              <span class="mo" id="MathJax-Span-977">
               +
              </span>
              <span class="msubsup" id="MathJax-Span-978">
               <span class="texatom" id="MathJax-Span-979">
                <span class="mrow" id="MathJax-Span-980">
                 <span class="mi" id="MathJax-Span-981">
                  ν
                 </span>
                </span>
               </span>
               <span class="mi" id="MathJax-Span-982">
                T
               </span>
              </span>
              <span class="mo" id="MathJax-Span-983">
               (
              </span>
              <span class="texatom" id="MathJax-Span-984">
               <span class="mrow" id="MathJax-Span-985">
                <span class="mi" id="MathJax-Span-986">
                 C
                </span>
                <span class="mi" id="MathJax-Span-987">
                 x
                </span>
               </span>
              </span>
              <span class="mo" id="MathJax-Span-988">
               −
              </span>
              <span class="texatom" id="MathJax-Span-989">
               <span class="mrow" id="MathJax-Span-990">
                <span class="mi" id="MathJax-Span-991">
                 d
                </span>
               </span>
              </span>
              <span class="mo" id="MathJax-Span-992">
               )
              </span>
              <span class="mo" id="MathJax-Span-993">
               )
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-996">
             <span class="mrow" id="MathJax-Span-997">
              <span class="mo" id="MathJax-Span-998">
               =
              </span>
              <span class="texatom" id="MathJax-Span-999">
               <span class="mrow" id="MathJax-Span-1000">
                <span class="mo" id="MathJax-Span-1001">
                 −
                </span>
                <span class="msubsup" id="MathJax-Span-1002">
                 <span class="mi" id="MathJax-Span-1003">
                  b
                 </span>
                 <span class="mi" id="MathJax-Span-1004">
                  T
                 </span>
                </span>
                <span class="mi" id="MathJax-Span-1005">
                 λ
                </span>
                <span class="mo" id="MathJax-Span-1006">
                 −
                </span>
                <span class="msubsup" id="MathJax-Span-1007">
                 <span class="mi" id="MathJax-Span-1008">
                  d
                 </span>
                 <span class="mi" id="MathJax-Span-1009">
                  T
                 </span>
                </span>
                <span class="mi" id="MathJax-Span-1010">
                 ν
                </span>
               </span>
              </span>
              <span class="mo" id="MathJax-Span-1011">
               +
              </span>
              <span class="munderover" id="MathJax-Span-1012">
               <span class="mo" id="MathJax-Span-1013">
                inf
               </span>
               <span class="texatom" id="MathJax-Span-1014">
                <span class="mrow" id="MathJax-Span-1015">
                 <span class="mi" id="MathJax-Span-1016">
                  x
                 </span>
                </span>
               </span>
              </span>
              <span class="mo" id="MathJax-Span-1017">
               (
              </span>
              <span class="msubsup" id="MathJax-Span-1018">
               <span class="mi" id="MathJax-Span-1019">
                f
               </span>
               <span class="mn" id="MathJax-Span-1020">
                0
               </span>
              </span>
              <span class="mo" id="MathJax-Span-1021">
               (
              </span>
              <span class="texatom" id="MathJax-Span-1022">
               <span class="mrow" id="MathJax-Span-1023">
                <span class="mi" id="MathJax-Span-1024">
                 x
                </span>
               </span>
              </span>
              <span class="mo" id="MathJax-Span-1025">
               )
              </span>
              <span class="mo" id="MathJax-Span-1026">
               +
              </span>
              <span class="mo" id="MathJax-Span-1027">
               (
              </span>
              <span class="texatom" id="MathJax-Span-1028">
               <span class="mrow" id="MathJax-Span-1029">
                <span class="msubsup" id="MathJax-Span-1030">
                 <span class="mi" id="MathJax-Span-1031">
                  A
                 </span>
                 <span class="mi" id="MathJax-Span-1032">
                  T
                 </span>
                </span>
                <span class="mi" id="MathJax-Span-1033">
                 λ
                </span>
                <span class="mo" id="MathJax-Span-1034">
                 +
                </span>
                <span class="msubsup" id="MathJax-Span-1035">
                 <span class="mi" id="MathJax-Span-1036">
                  C
                 </span>
                 <span class="mi" id="MathJax-Span-1037">
                  T
                 </span>
                </span>
                <span class="mi" id="MathJax-Span-1038">
                 ν
                </span>
               </span>
              </span>
              <span class="msubsup" id="MathJax-Span-1039">
               <span class="mo" id="MathJax-Span-1040">
                )
               </span>
               <span class="mi" id="MathJax-Span-1041">
                T
               </span>
              </span>
              <span class="texatom" id="MathJax-Span-1042">
               <span class="mrow" id="MathJax-Span-1043">
                <span class="mi" id="MathJax-Span-1044">
                 x
                </span>
               </span>
              </span>
              <span class="mo" id="MathJax-Span-1045">
               )
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1048">
             <span class="mrow" id="MathJax-Span-1049">
              <span class="mo" id="MathJax-Span-1050">
               =
              </span>
              <span class="texatom" id="MathJax-Span-1051">
               <span class="mrow" id="MathJax-Span-1052">
                <span class="mo" id="MathJax-Span-1053">
                 −
                </span>
                <span class="msubsup" id="MathJax-Span-1054">
                 <span class="mi" id="MathJax-Span-1055">
                  b
                 </span>
                 <span class="mi" id="MathJax-Span-1056">
                  T
                 </span>
                </span>
                <span class="mi" id="MathJax-Span-1057">
                 λ
                </span>
                <span class="mo" id="MathJax-Span-1058">
                 −
                </span>
                <span class="msubsup" id="MathJax-Span-1059">
                 <span class="mi" id="MathJax-Span-1060">
                  d
                 </span>
                 <span class="mi" id="MathJax-Span-1061">
                  T
                 </span>
                </span>
                <span class="mi" id="MathJax-Span-1062">
                 ν
                </span>
               </span>
              </span>
              <span class="mo" id="MathJax-Span-1063">
               −
              </span>
              <span class="msubsup" id="MathJax-Span-1064">
               <span class="mi" id="MathJax-Span-1065">
                f
               </span>
               <span class="mo" id="MathJax-Span-1066">
                ⋆
               </span>
               <span class="mn" id="MathJax-Span-1067">
                0
               </span>
              </span>
              <span class="mo" id="MathJax-Span-1068">
               (
              </span>
              <span class="texatom" id="MathJax-Span-1069">
               <span class="mrow" id="MathJax-Span-1070">
                <span class="mo" id="MathJax-Span-1071">
                 −
                </span>
                <span class="msubsup" id="MathJax-Span-1072">
                 <span class="mi" id="MathJax-Span-1073">
                  A
                 </span>
                 <span class="mi" id="MathJax-Span-1074">
                  T
                 </span>
                </span>
                <span class="mi" id="MathJax-Span-1075">
                 λ
                </span>
                <span class="mo" id="MathJax-Span-1076">
                 −
                </span>
                <span class="msubsup" id="MathJax-Span-1077">
                 <span class="mi" id="MathJax-Span-1078">
                  C
                 </span>
                 <span class="mi" id="MathJax-Span-1079">
                  T
                 </span>
                </span>
                <span class="mi" id="MathJax-Span-1080">
                 ν
                </span>
               </span>
              </span>
              <span class="mo" id="MathJax-Span-1081">
               )
              </span>
             </span>
            </span>
           </span>
           <span class="mtext" id="MathJax-Span-1082">
           </span>
          </span>
         </span>
        </span>
       </div>
       <p>
        对偶函数的定义域由共轭函数
        <span class="editor-math-element" data-id="JTI1MjRmJTI1NUUlMjU1Q3N0YXIlMjUyNA==">
         <span class="MathJax" id="MathJax-Element-60-Frame">
          <span class="math" id="MathJax-Span-1083">
           <span class="mrow" id="MathJax-Span-1084">
            <span class="msubsup" id="MathJax-Span-1085">
             <span class="mi" id="MathJax-Span-1086">
              f
             </span>
             <span class="mo" id="MathJax-Span-1087">
              ⋆
             </span>
            </span>
           </span>
          </span>
         </span>
        </span>
        的定义域所确定：
       </p>
       <div class="MathJax_Display">
        <span class="MathJax" id="MathJax-Element-61-Frame">
         <span class="math" id="MathJax-Span-1088">
          <span class="mrow" id="MathJax-Span-1089">
           <span class="texatom" id="MathJax-Span-1090">
            <span class="mrow" id="MathJax-Span-1091">
             <span class="mtext" id="MathJax-Span-1092">
              dom
             </span>
            </span>
           </span>
           <span class="mi" id="MathJax-Span-1093">
            g
           </span>
           <span class="mo" id="MathJax-Span-1094">
            =
           </span>
           <span class="mrow" id="MathJax-Span-1095">
            <span class="mo" id="MathJax-Span-1096">
             {
            </span>
            <span class="mo" id="MathJax-Span-1097">
             (
            </span>
            <span class="mi" id="MathJax-Span-1098">
             λ
            </span>
            <span class="mo" id="MathJax-Span-1099">
             ,
            </span>
            <span class="mi" id="MathJax-Span-1100">
             ν
            </span>
            <span class="mo" id="MathJax-Span-1101">
             )
            </span>
            <span class="mo" id="MathJax-Span-1102">
             ∣
            </span>
            <span class="mo" id="MathJax-Span-1103">
             −
            </span>
            <span class="msubsup" id="MathJax-Span-1104">
             <span class="mi" id="MathJax-Span-1105">
              A
             </span>
             <span class="mi" id="MathJax-Span-1106">
              T
             </span>
            </span>
            <span class="mi" id="MathJax-Span-1107">
             λ
            </span>
            <span class="mo" id="MathJax-Span-1108">
             −
            </span>
            <span class="msubsup" id="MathJax-Span-1109">
             <span class="mi" id="MathJax-Span-1110">
              C
             </span>
             <span class="mi" id="MathJax-Span-1111">
              T
             </span>
            </span>
            <span class="mi" id="MathJax-Span-1112">
             ν
            </span>
            <span class="mo" id="MathJax-Span-1113">
             ∈
            </span>
            <span class="texatom" id="MathJax-Span-1114">
             <span class="mrow" id="MathJax-Span-1115">
              <span class="mtext" id="MathJax-Span-1116">
               dom
              </span>
             </span>
            </span>
            <span class="msubsup" id="MathJax-Span-1117">
             <span class="mi" id="MathJax-Span-1118">
              f
             </span>
             <span class="mo" id="MathJax-Span-1119">
              ∗
             </span>
             <span class="mn" id="MathJax-Span-1120">
              0
             </span>
            </span>
            <span class="mo" id="MathJax-Span-1121">
             }
            </span>
           </span>
           <span class="mtext" id="MathJax-Span-1122">
           </span>
          </span>
         </span>
        </span>
       </div>
       <h2>
        拉格朗日对偶问题
        <span id="anchor_JTI1dTYyQzklMjV1NjgzQyUyNXU2NzE3JTI1dTY1RTUlMjV1NUJGOSUyNXU1MDc2JTI1dTk1RUUlMjV1OTg5ODY=">
        </span>
       </h2>
       <p>
        由于我们知道
        <span class="editor-math-element" data-id="JTI1MjRnJTI1MjglMjU3QiUyNTVDYmYlMjUyMCUyNTVDbGFtYmRhJTI1N0QlMjUyQyUyNTdCJTI1NUNiZiUyNTIwJTI1NUNudSUyNTdEJTI1MjklMjUyNA==">
         <span class="MathJax" id="MathJax-Element-62-Frame">
          <span class="math" id="MathJax-Span-1123">
           <span class="mrow" id="MathJax-Span-1124">
            <span class="mi" id="MathJax-Span-1125">
             g
            </span>
            <span class="mo" id="MathJax-Span-1126">
             (
            </span>
            <span class="texatom" id="MathJax-Span-1127">
             <span class="mrow" id="MathJax-Span-1128">
              <span class="mi" id="MathJax-Span-1129">
               λ
              </span>
             </span>
            </span>
            <span class="mo" id="MathJax-Span-1130">
             ,
            </span>
            <span class="texatom" id="MathJax-Span-1131">
             <span class="mrow" id="MathJax-Span-1132">
              <span class="mi" id="MathJax-Span-1133">
               ν
              </span>
             </span>
            </span>
            <span class="mo" id="MathJax-Span-1134">
             )
            </span>
           </span>
          </span>
         </span>
        </span>
        是一定不会大于原问题的最优解的，我们可以通过构造一个如下的最优化问题来寻找原始优化问题的最优下界：
       </p>
       <div class="MathJax_Display">
        <span class="MathJax" id="MathJax-Element-63-Frame">
         <span class="math" id="MathJax-Span-1135">
          <span class="mrow" id="MathJax-Span-1136">
           <span class="mtable" id="MathJax-Span-1137">
            <span class="mtd" id="MathJax-Span-1138">
             <span class="mrow" id="MathJax-Span-1139">
              <span class="mtext" id="MathJax-Span-1140">
               maximize
              </span>
              <span class="mspace" id="MathJax-Span-1141">
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1154">
             <span class="mrow" id="MathJax-Span-1155">
              <span class="mtext" id="MathJax-Span-1156">
               subject to
              </span>
              <span class="mspace" id="MathJax-Span-1157">
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1142">
             <span class="mrow" id="MathJax-Span-1143">
              <span class="mi" id="MathJax-Span-1144">
               g
              </span>
              <span class="mo" id="MathJax-Span-1145">
               (
              </span>
              <span class="texatom" id="MathJax-Span-1146">
               <span class="mrow" id="MathJax-Span-1147">
                <span class="mi" id="MathJax-Span-1148">
                 λ
                </span>
               </span>
              </span>
              <span class="mo" id="MathJax-Span-1149">
               ,
              </span>
              <span class="texatom" id="MathJax-Span-1150">
               <span class="mrow" id="MathJax-Span-1151">
                <span class="mi" id="MathJax-Span-1152">
                 ν
                </span>
               </span>
              </span>
              <span class="mo" id="MathJax-Span-1153">
               )
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1158">
             <span class="mrow" id="MathJax-Span-1159">
              <span class="texatom" id="MathJax-Span-1160">
               <span class="mrow" id="MathJax-Span-1161">
                <span class="mi" id="MathJax-Span-1162">
                 λ
                </span>
               </span>
              </span>
              <span class="mo" id="MathJax-Span-1163">
               ≥
              </span>
              <span class="mn" id="MathJax-Span-1164">
               0
              </span>
             </span>
            </span>
           </span>
           <span class="mtext" id="MathJax-Span-1165">
           </span>
          </span>
         </span>
        </span>
       </div>
       <p>
        我们用
        <span class="editor-math-element" data-id="JTI1MjQlMjUyOCUyNTVDbGFtYmRhJTI1NUUlMjU1Q3N0YXIlMjUyQyUyNTVDbnUlMjU1RSUyNTVDc3RhciUyNTI5JTI1MjQ=">
         <span class="MathJax" id="MathJax-Element-64-Frame">
          <span class="math" id="MathJax-Span-1166">
           <span class="mrow" id="MathJax-Span-1167">
            <span class="mo" id="MathJax-Span-1168">
             (
            </span>
            <span class="msubsup" id="MathJax-Span-1169">
             <span class="mi" id="MathJax-Span-1170">
              λ
             </span>
             <span class="mo" id="MathJax-Span-1171">
              ⋆
             </span>
            </span>
            <span class="mo" id="MathJax-Span-1172">
             ,
            </span>
            <span class="msubsup" id="MathJax-Span-1173">
             <span class="mi" id="MathJax-Span-1174">
              ν
             </span>
             <span class="mo" id="MathJax-Span-1175">
              ⋆
             </span>
            </span>
            <span class="mo" id="MathJax-Span-1176">
             )
            </span>
           </span>
          </span>
         </span>
        </span>
        来代表这个问题找到最优时候的对应的拉格朗日乘子的值。
       </p>
       <p>
        由于我们知道拉格朗日对偶函数
        <span class="editor-math-element" data-id="JTI1MjRnJTI1MjglMjU3QiUyNTVDYmYlMjUyMCUyNTVDbGFtYmRhJTI1N0QlMjUyQyUyNTdCJTI1NUNiZiUyNTIwJTI1NUNudSUyNTdEJTI1MjklMjUyNA==">
         <span class="MathJax" id="MathJax-Element-65-Frame">
          <span class="math" id="MathJax-Span-1177">
           <span class="mrow" id="MathJax-Span-1178">
            <span class="mi" id="MathJax-Span-1179">
             g
            </span>
            <span class="mo" id="MathJax-Span-1180">
             (
            </span>
            <span class="texatom" id="MathJax-Span-1181">
             <span class="mrow" id="MathJax-Span-1182">
              <span class="mi" id="MathJax-Span-1183">
               λ
              </span>
             </span>
            </span>
            <span class="mo" id="MathJax-Span-1184">
             ,
            </span>
            <span class="texatom" id="MathJax-Span-1185">
             <span class="mrow" id="MathJax-Span-1186">
              <span class="mi" id="MathJax-Span-1187">
               ν
              </span>
             </span>
            </span>
            <span class="mo" id="MathJax-Span-1188">
             )
            </span>
           </span>
          </span>
         </span>
        </span>
        一定是凹的（不论原始优化问题凹凸性如何），因此我们知道这个拉格朗日对偶问题一定是个凸优化问题。
       </p>
       <h3>
        如何显式的表述拉格朗日对偶问题
        <span id="anchor_JTI1dTU5ODIlMjV1NEY1NSUyNXU2NjNFJTI1dTVGMEYlMjV1NzY4NCUyNXU4ODY4JTI1dThGRjAlMjV1NjJDOSUyNXU2ODNDJTI1dTY3MTclMjV1NjVFNSUyNXU1QkY5JTI1dTUwNzYlMjV1OTVFRSUyNXU5ODk4Nw==">
        </span>
       </h3>
       <p>
        上面形式的拉格朗日对偶问题很难在实际中求解。通常情况下为了求解，我们需要一些更明确的条件来把拉格朗日对偶问题表述出来。一般我们如下几种方法。
       </p>
       <h4>
        由定义消去下确界
        <span id="anchor_JTI1dTc1MzElMjV1NUI5QSUyNXU0RTQ5JTI1dTZEODglMjV1NTNCQiUyNXU0RTBCJTI1dTc4NkUlMjV1NzU0Qzg=">
        </span>
       </h4>
       <p>
        如果拉格朗日函数能够简单的求得下确界。我们就可以直接消去原始问题的变量，得到明确的对偶问题。
       </p>
       <p>
        例如对于：
       </p>
       <div class="MathJax_Display">
        <span class="MathJax" id="MathJax-Element-66-Frame">
         <span class="math" id="MathJax-Span-1189">
          <span class="mrow" id="MathJax-Span-1190">
           <span class="mtable" id="MathJax-Span-1191">
            <span class="mtd" id="MathJax-Span-1192">
             <span class="mrow" id="MathJax-Span-1193">
              <span class="mtext" id="MathJax-Span-1194">
               minimize
              </span>
              <span class="mspace" id="MathJax-Span-1195">
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1202">
             <span class="mrow" id="MathJax-Span-1203">
              <span class="mtext" id="MathJax-Span-1204">
               subject to
              </span>
              <span class="mspace" id="MathJax-Span-1205">
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1196">
             <span class="mrow" id="MathJax-Span-1197">
              <span class="msubsup" id="MathJax-Span-1198">
               <span class="mi" id="MathJax-Span-1199">
                x
               </span>
               <span class="mi" id="MathJax-Span-1200">
                T
               </span>
              </span>
              <span class="mi" id="MathJax-Span-1201">
               x
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1206">
             <span class="mrow" id="MathJax-Span-1207">
              <span class="mi" id="MathJax-Span-1208">
               A
              </span>
              <span class="mi" id="MathJax-Span-1209">
               x
              </span>
              <span class="mo" id="MathJax-Span-1210">
               =
              </span>
              <span class="mi" id="MathJax-Span-1211">
               b
              </span>
             </span>
            </span>
           </span>
           <span class="mtext" id="MathJax-Span-1212">
           </span>
          </span>
         </span>
        </span>
       </div>
       <p>
        拉格朗日函数为
        <span class="editor-math-element" data-id="JTI1MjRMJTI1Mjh4JTI1MkMlMjU1Q251JTI1MjklMjUyMCUyNTNEJTI1MjB4JTI1NUVUJTI1MjB4JTI1MjAlMkIlMjUyMCUyNTVDbnUlMjU1RVQlMjUyMCUyNTI4QXgtYiUyNTI5JTI1MjQ=">
         <span class="MathJax" id="MathJax-Element-67-Frame">
          <span class="math" id="MathJax-Span-1213">
           <span class="mrow" id="MathJax-Span-1214">
            <span class="mi" id="MathJax-Span-1215">
             L
            </span>
            <span class="mo" id="MathJax-Span-1216">
             (
            </span>
            <span class="mi" id="MathJax-Span-1217">
             x
            </span>
            <span class="mo" id="MathJax-Span-1218">
             ,
            </span>
            <span class="mi" id="MathJax-Span-1219">
             ν
            </span>
            <span class="mo" id="MathJax-Span-1220">
             )
            </span>
            <span class="mo" id="MathJax-Span-1221">
             =
            </span>
            <span class="msubsup" id="MathJax-Span-1222">
             <span class="mi" id="MathJax-Span-1223">
              x
             </span>
             <span class="mi" id="MathJax-Span-1224">
              T
             </span>
            </span>
            <span class="mi" id="MathJax-Span-1225">
             x
            </span>
            <span class="mo" id="MathJax-Span-1226">
             +
            </span>
            <span class="msubsup" id="MathJax-Span-1227">
             <span class="mi" id="MathJax-Span-1228">
              ν
             </span>
             <span class="mi" id="MathJax-Span-1229">
              T
             </span>
            </span>
            <span class="mo" id="MathJax-Span-1230">
             (
            </span>
            <span class="mi" id="MathJax-Span-1231">
             A
            </span>
            <span class="mi" id="MathJax-Span-1232">
             x
            </span>
            <span class="mo" id="MathJax-Span-1233">
             −
            </span>
            <span class="mi" id="MathJax-Span-1234">
             b
            </span>
            <span class="mo" id="MathJax-Span-1235">
             )
            </span>
           </span>
          </span>
         </span>
        </span>
        。这显然是个凸函数，我们可以直接求极值，令其导数为0：
       </p>
       <div class="MathJax_Display">
        <span class="MathJax" id="MathJax-Element-68-Frame">
         <span class="math" id="MathJax-Span-1236">
          <span class="mrow" id="MathJax-Span-1237">
           <span class="msubsup" id="MathJax-Span-1238">
            <span class="mi" id="MathJax-Span-1239">
             ∇
            </span>
            <span class="texatom" id="MathJax-Span-1240">
             <span class="mrow" id="MathJax-Span-1241">
              <span class="mi" id="MathJax-Span-1242">
               x
              </span>
             </span>
            </span>
           </span>
           <span class="mi" id="MathJax-Span-1243">
            L
           </span>
           <span class="mo" id="MathJax-Span-1244">
            (
           </span>
           <span class="mi" id="MathJax-Span-1245">
            x
           </span>
           <span class="mo" id="MathJax-Span-1246">
            ,
           </span>
           <span class="mi" id="MathJax-Span-1247">
            ν
           </span>
           <span class="mo" id="MathJax-Span-1248">
            )
           </span>
           <span class="mo" id="MathJax-Span-1249">
            =
           </span>
           <span class="mn" id="MathJax-Span-1250">
            2
           </span>
           <span class="mi" id="MathJax-Span-1251">
            x
           </span>
           <span class="mo" id="MathJax-Span-1252">
            +
           </span>
           <span class="msubsup" id="MathJax-Span-1253">
            <span class="mi" id="MathJax-Span-1254">
             A
            </span>
            <span class="mi" id="MathJax-Span-1255">
             T
            </span>
           </span>
           <span class="mi" id="MathJax-Span-1256">
            ν
           </span>
           <span class="mo" id="MathJax-Span-1257">
            =
           </span>
           <span class="mn" id="MathJax-Span-1258">
            0
           </span>
           <span class="mtext" id="MathJax-Span-1259">
           </span>
          </span>
         </span>
        </span>
       </div>
       <p>
        可得
        <span class="editor-math-element" data-id="JTI1MjR4JTI1M0QtJTI1MjgxJTJGMiUyNTI5QSUyNTVFVCUyNTIwJTI1NUNudSUyNTI0">
         <span class="MathJax" id="MathJax-Element-69-Frame">
          <span class="math" id="MathJax-Span-1260">
           <span class="mrow" id="MathJax-Span-1261">
            <span class="mi" id="MathJax-Span-1262">
             x
            </span>
            <span class="mo" id="MathJax-Span-1263">
             =
            </span>
            <span class="mo" id="MathJax-Span-1264">
             −
            </span>
            <span class="mo" id="MathJax-Span-1265">
             (
            </span>
            <span class="mn" id="MathJax-Span-1266">
             1
            </span>
            <span class="texatom" id="MathJax-Span-1267">
             <span class="mrow" id="MathJax-Span-1268">
              <span class="mo" id="MathJax-Span-1269">
               /
              </span>
             </span>
            </span>
            <span class="mn" id="MathJax-Span-1270">
             2
            </span>
            <span class="mo" id="MathJax-Span-1271">
             )
            </span>
            <span class="msubsup" id="MathJax-Span-1272">
             <span class="mi" id="MathJax-Span-1273">
              A
             </span>
             <span class="mi" id="MathJax-Span-1274">
              T
             </span>
            </span>
            <span class="mi" id="MathJax-Span-1275">
             ν
            </span>
           </span>
          </span>
         </span>
        </span>
        ，带入拉格朗日函数：
       </p>
       <div class="MathJax_Display">
        <span class="MathJax" id="MathJax-Element-70-Frame">
         <span class="math" id="MathJax-Span-1276">
          <span class="mrow" id="MathJax-Span-1277">
           <span class="mi" id="MathJax-Span-1278">
            g
           </span>
           <span class="mo" id="MathJax-Span-1279">
            (
           </span>
           <span class="mi" id="MathJax-Span-1280">
            ν
           </span>
           <span class="mo" id="MathJax-Span-1281">
            )
           </span>
           <span class="mo" id="MathJax-Span-1282">
            =
           </span>
           <span class="mi" id="MathJax-Span-1283">
            L
           </span>
           <span class="mo" id="MathJax-Span-1284">
            (
           </span>
           <span class="mo" id="MathJax-Span-1285">
            −
           </span>
           <span class="mo" id="MathJax-Span-1286">
            (
           </span>
           <span class="mn" id="MathJax-Span-1287">
            1
           </span>
           <span class="texatom" id="MathJax-Span-1288">
            <span class="mrow" id="MathJax-Span-1289">
             <span class="mo" id="MathJax-Span-1290">
              /
             </span>
            </span>
           </span>
           <span class="mn" id="MathJax-Span-1291">
            2
           </span>
           <span class="mo" id="MathJax-Span-1292">
            )
           </span>
           <span class="msubsup" id="MathJax-Span-1293">
            <span class="mi" id="MathJax-Span-1294">
             A
            </span>
            <span class="mi" id="MathJax-Span-1295">
             T
            </span>
           </span>
           <span class="mi" id="MathJax-Span-1296">
            ν
           </span>
           <span class="mo" id="MathJax-Span-1297">
            ,
           </span>
           <span class="mi" id="MathJax-Span-1298">
            ν
           </span>
           <span class="mo" id="MathJax-Span-1299">
            )
           </span>
           <span class="mo" id="MathJax-Span-1300">
            =
           </span>
           <span class="mo" id="MathJax-Span-1301">
            −
           </span>
           <span class="mo" id="MathJax-Span-1302">
            (
           </span>
           <span class="mn" id="MathJax-Span-1303">
            1
           </span>
           <span class="texatom" id="MathJax-Span-1304">
            <span class="mrow" id="MathJax-Span-1305">
             <span class="mo" id="MathJax-Span-1306">
              /
             </span>
            </span>
           </span>
           <span class="mn" id="MathJax-Span-1307">
            4
           </span>
           <span class="mo" id="MathJax-Span-1308">
            )
           </span>
           <span class="msubsup" id="MathJax-Span-1309">
            <span class="mi" id="MathJax-Span-1310">
             ν
            </span>
            <span class="mi" id="MathJax-Span-1311">
             T
            </span>
           </span>
           <span class="mi" id="MathJax-Span-1312">
            A
           </span>
           <span class="msubsup" id="MathJax-Span-1313">
            <span class="mi" id="MathJax-Span-1314">
             A
            </span>
            <span class="mi" id="MathJax-Span-1315">
             T
            </span>
           </span>
           <span class="mi" id="MathJax-Span-1316">
            ν
           </span>
           <span class="mo" id="MathJax-Span-1317">
            −
           </span>
           <span class="msubsup" id="MathJax-Span-1318">
            <span class="mi" id="MathJax-Span-1319">
             b
            </span>
            <span class="mi" id="MathJax-Span-1320">
             T
            </span>
           </span>
           <span class="mi" id="MathJax-Span-1321">
            ν
           </span>
           <span class="mtext" id="MathJax-Span-1322">
           </span>
          </span>
         </span>
        </span>
       </div>
       <p>
        可以看到，这个问题的对偶问题变成了一个无约束的优化问题：
       </p>
       <div class="MathJax_Display">
        <span class="MathJax" id="MathJax-Element-71-Frame">
         <span class="math" id="MathJax-Span-1323">
          <span class="mrow" id="MathJax-Span-1324">
           <span class="mtext" id="MathJax-Span-1325">
            maximize
           </span>
           <span class="mspace" id="MathJax-Span-1326">
           </span>
           <span class="mo" id="MathJax-Span-1327">
            −
           </span>
           <span class="mo" id="MathJax-Span-1328">
            (
           </span>
           <span class="mn" id="MathJax-Span-1329">
            1
           </span>
           <span class="texatom" id="MathJax-Span-1330">
            <span class="mrow" id="MathJax-Span-1331">
             <span class="mo" id="MathJax-Span-1332">
              /
             </span>
            </span>
           </span>
           <span class="mn" id="MathJax-Span-1333">
            4
           </span>
           <span class="mo" id="MathJax-Span-1334">
            )
           </span>
           <span class="msubsup" id="MathJax-Span-1335">
            <span class="mi" id="MathJax-Span-1336">
             ν
            </span>
            <span class="mi" id="MathJax-Span-1337">
             T
            </span>
           </span>
           <span class="mi" id="MathJax-Span-1338">
            A
           </span>
           <span class="msubsup" id="MathJax-Span-1339">
            <span class="mi" id="MathJax-Span-1340">
             A
            </span>
            <span class="mi" id="MathJax-Span-1341">
             T
            </span>
           </span>
           <span class="mi" id="MathJax-Span-1342">
            ν
           </span>
           <span class="mo" id="MathJax-Span-1343">
            −
           </span>
           <span class="msubsup" id="MathJax-Span-1344">
            <span class="mi" id="MathJax-Span-1345">
             b
            </span>
            <span class="mi" id="MathJax-Span-1346">
             T
            </span>
           </span>
           <span class="mi" id="MathJax-Span-1347">
            ν
           </span>
           <span class="mtext" id="MathJax-Span-1348">
           </span>
          </span>
         </span>
        </span>
       </div>
       <h4>
        隐式求解约束
        <span id="anchor_JTI1dTk2OTAlMjV1NUYwRiUyNXU2QzQyJTI1dTg5RTMlMjV1N0VBNiUyNXU2NzVGOQ==">
        </span>
       </h4>
       <p>
        有时候拉格朗日对偶函数可以取到无穷。为了得到有意义的解，我们可以求出对偶可行域，即让
        <span class="editor-math-element" data-id="JTI1MjRnJTI1MjglMjU1Q2xhbWJkYSUyNTJDJTI1NUNudSUyNTI5JTI1MjAlMjUzRCUyNTIwJTI1NUNpbmZfJTI1N0J4JTI1NUNpbiUyNTVDbWF0aGNhbCUyNTdCRCUyNTdEJTI1N0QlMjUyMEwlMjUyOHglMjUyQyUyNTVDbGFtYmRhJTI1MkMlMjU1Q251JTI1MjklMjUyMCUyNTVDZ3QlMjUyMC0lMjU1Q2luZnR5JTI1MjQ=">
         <span class="MathJax" id="MathJax-Element-72-Frame">
          <span class="math" id="MathJax-Span-1349">
           <span class="mrow" id="MathJax-Span-1350">
            <span class="mi" id="MathJax-Span-1351">
             g
            </span>
            <span class="mo" id="MathJax-Span-1352">
             (
            </span>
            <span class="mi" id="MathJax-Span-1353">
             λ
            </span>
            <span class="mo" id="MathJax-Span-1354">
             ,
            </span>
            <span class="mi" id="MathJax-Span-1355">
             ν
            </span>
            <span class="mo" id="MathJax-Span-1356">
             )
            </span>
            <span class="mo" id="MathJax-Span-1357">
             =
            </span>
            <span class="munderover" id="MathJax-Span-1358">
             <span class="mo" id="MathJax-Span-1359">
              inf
             </span>
             <span class="texatom" id="MathJax-Span-1360">
              <span class="mrow" id="MathJax-Span-1361">
               <span class="mi" id="MathJax-Span-1362">
                x
               </span>
               <span class="mo" id="MathJax-Span-1363">
                ∈
               </span>
               <span class="texatom" id="MathJax-Span-1364">
                <span class="mrow" id="MathJax-Span-1365">
                 <span class="mi" id="MathJax-Span-1366">
                  D
                 </span>
                </span>
               </span>
              </span>
             </span>
            </span>
            <span class="mi" id="MathJax-Span-1367">
             L
            </span>
            <span class="mo" id="MathJax-Span-1368">
             (
            </span>
            <span class="mi" id="MathJax-Span-1369">
             x
            </span>
            <span class="mo" id="MathJax-Span-1370">
             ,
            </span>
            <span class="mi" id="MathJax-Span-1371">
             λ
            </span>
            <span class="mo" id="MathJax-Span-1372">
             ,
            </span>
            <span class="mi" id="MathJax-Span-1373">
             ν
            </span>
            <span class="mo" id="MathJax-Span-1374">
             )
            </span>
            <span class="mo" id="MathJax-Span-1375">
             &gt;
            </span>
            <span class="mo" id="MathJax-Span-1376">
             −
            </span>
            <span class="mi" id="MathJax-Span-1377">
             ∞
            </span>
           </span>
          </span>
         </span>
        </span>
        成立所需要的约束。然后可得对偶问题。
       </p>
       <p>
        例如对于标准线性规划问题：
       </p>
       <div class="MathJax_Display">
        <span class="MathJax" id="MathJax-Element-73-Frame">
         <span class="math" id="MathJax-Span-1378">
          <span class="mrow" id="MathJax-Span-1379">
           <span class="mtable" id="MathJax-Span-1380">
            <span class="mtd" id="MathJax-Span-1381">
             <span class="mrow" id="MathJax-Span-1382">
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1394">
             <span class="mrow" id="MathJax-Span-1395">
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1407">
             <span class="mrow" id="MathJax-Span-1408">
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1383">
             <span class="mrow" id="MathJax-Span-1384">
              <span class="mtext" id="MathJax-Span-1385">
               minimize
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1396">
             <span class="mrow" id="MathJax-Span-1397">
              <span class="mtext" id="MathJax-Span-1398">
               subject to
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1409">
             <span class="mrow" id="MathJax-Span-1410">
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1386">
             <span class="mrow" id="MathJax-Span-1387">
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1399">
             <span class="mrow" id="MathJax-Span-1400">
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1411">
             <span class="mrow" id="MathJax-Span-1412">
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1388">
             <span class="mrow" id="MathJax-Span-1389">
              <span class="msubsup" id="MathJax-Span-1390">
               <span class="mi" id="MathJax-Span-1391">
                c
               </span>
               <span class="mi" id="MathJax-Span-1392">
                T
               </span>
              </span>
              <span class="mi" id="MathJax-Span-1393">
               x
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1401">
             <span class="mrow" id="MathJax-Span-1402">
              <span class="mi" id="MathJax-Span-1403">
               A
              </span>
              <span class="mi" id="MathJax-Span-1404">
               x
              </span>
              <span class="mo" id="MathJax-Span-1405">
               =
              </span>
              <span class="mi" id="MathJax-Span-1406">
               b
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1413">
             <span class="mrow" id="MathJax-Span-1414">
              <span class="mi" id="MathJax-Span-1415">
               x
              </span>
              <span class="mo" id="MathJax-Span-1416">
               ⪰
              </span>
              <span class="mn" id="MathJax-Span-1417">
               0
              </span>
             </span>
            </span>
           </span>
           <span class="mtext" id="MathJax-Span-1418">
           </span>
          </span>
         </span>
        </span>
       </div>
       <p>
        拉格朗日函数为：
       </p>
       <div class="MathJax_Display">
        <span class="MathJax" id="MathJax-Element-74-Frame">
         <span class="math" id="MathJax-Span-1419">
          <span class="mrow" id="MathJax-Span-1420">
           <span class="mi" id="MathJax-Span-1421">
            L
           </span>
           <span class="mo" id="MathJax-Span-1422">
            (
           </span>
           <span class="mi" id="MathJax-Span-1423">
            x
           </span>
           <span class="mo" id="MathJax-Span-1424">
            ,
           </span>
           <span class="mi" id="MathJax-Span-1425">
            ν
           </span>
           <span class="mo" id="MathJax-Span-1426">
            )
           </span>
           <span class="mo" id="MathJax-Span-1427">
            =
           </span>
           <span class="msubsup" id="MathJax-Span-1428">
            <span class="mi" id="MathJax-Span-1429">
             c
            </span>
            <span class="mi" id="MathJax-Span-1430">
             T
            </span>
           </span>
           <span class="mi" id="MathJax-Span-1431">
            x
           </span>
           <span class="mo" id="MathJax-Span-1432">
            −
           </span>
           <span class="munderover" id="MathJax-Span-1433">
            <span class="mo" id="MathJax-Span-1434">
             ∑
            </span>
            <span class="texatom" id="MathJax-Span-1435">
             <span class="mrow" id="MathJax-Span-1436">
              <span class="mi" id="MathJax-Span-1437">
               i
              </span>
              <span class="mo" id="MathJax-Span-1438">
               =
              </span>
              <span class="mn" id="MathJax-Span-1439">
               1
              </span>
             </span>
            </span>
            <span class="mi" id="MathJax-Span-1440">
             m
            </span>
           </span>
           <span class="msubsup" id="MathJax-Span-1441">
            <span class="mi" id="MathJax-Span-1442">
             λ
            </span>
            <span class="mi" id="MathJax-Span-1443">
             i
            </span>
           </span>
           <span class="msubsup" id="MathJax-Span-1444">
            <span class="mi" id="MathJax-Span-1445">
             x
            </span>
            <span class="mi" id="MathJax-Span-1446">
             i
            </span>
           </span>
           <span class="mo" id="MathJax-Span-1447">
            +
           </span>
           <span class="msubsup" id="MathJax-Span-1448">
            <span class="mi" id="MathJax-Span-1449">
             ν
            </span>
            <span class="mi" id="MathJax-Span-1450">
             T
            </span>
           </span>
           <span class="mo" id="MathJax-Span-1451">
            (
           </span>
           <span class="mi" id="MathJax-Span-1452">
            A
           </span>
           <span class="mi" id="MathJax-Span-1453">
            x
           </span>
           <span class="mo" id="MathJax-Span-1454">
            −
           </span>
           <span class="mi" id="MathJax-Span-1455">
            b
           </span>
           <span class="mo" id="MathJax-Span-1456">
            )
           </span>
           <span class="mo" id="MathJax-Span-1457">
            =
           </span>
           <span class="mo" id="MathJax-Span-1458">
            −
           </span>
           <span class="msubsup" id="MathJax-Span-1459">
            <span class="mi" id="MathJax-Span-1460">
             b
            </span>
            <span class="mi" id="MathJax-Span-1461">
             T
            </span>
           </span>
           <span class="mi" id="MathJax-Span-1462">
            ν
           </span>
           <span class="mo" id="MathJax-Span-1463">
            +
           </span>
           <span class="mo" id="MathJax-Span-1464">
            (
           </span>
           <span class="msubsup" id="MathJax-Span-1465">
            <span class="mi" id="MathJax-Span-1466">
             A
            </span>
            <span class="mi" id="MathJax-Span-1467">
             T
            </span>
           </span>
           <span class="mi" id="MathJax-Span-1468">
            ν
           </span>
           <span class="mo" id="MathJax-Span-1469">
            +
           </span>
           <span class="mi" id="MathJax-Span-1470">
            c
           </span>
           <span class="mo" id="MathJax-Span-1471">
            −
           </span>
           <span class="mi" id="MathJax-Span-1472">
            λ
           </span>
           <span class="msubsup" id="MathJax-Span-1473">
            <span class="mo" id="MathJax-Span-1474">
             )
            </span>
            <span class="mi" id="MathJax-Span-1475">
             T
            </span>
           </span>
           <span class="mi" id="MathJax-Span-1476">
            x
           </span>
           <span class="mtext" id="MathJax-Span-1477">
           </span>
          </span>
         </span>
        </span>
       </div>
       <p>
        应用下确界运算，可以得到：
       </p>
       <div class="MathJax_Display">
        <span class="MathJax" id="MathJax-Element-75-Frame">
         <span class="math" id="MathJax-Span-1478">
          <span class="mrow" id="MathJax-Span-1479">
           <span class="mi" id="MathJax-Span-1480">
            g
           </span>
           <span class="mo" id="MathJax-Span-1481">
            (
           </span>
           <span class="mi" id="MathJax-Span-1482">
            λ
           </span>
           <span class="mo" id="MathJax-Span-1483">
            ,
           </span>
           <span class="mi" id="MathJax-Span-1484">
            ν
           </span>
           <span class="mo" id="MathJax-Span-1485">
            )
           </span>
           <span class="mo" id="MathJax-Span-1486">
            =
           </span>
           <span class="mrow" id="MathJax-Span-1487">
            <span class="mo" id="MathJax-Span-1488">
             {
            </span>
            <span class="mtable" id="MathJax-Span-1489">
             <span class="mtd" id="MathJax-Span-1490">
              <span class="mrow" id="MathJax-Span-1491">
               <span class="mo" id="MathJax-Span-1492">
                −
               </span>
               <span class="msubsup" id="MathJax-Span-1493">
                <span class="mi" id="MathJax-Span-1494">
                 b
                </span>
                <span class="mi" id="MathJax-Span-1495">
                 T
                </span>
               </span>
               <span class="mi" id="MathJax-Span-1496">
                ν
               </span>
              </span>
             </span>
             <span class="mtd" id="MathJax-Span-1509">
              <span class="mrow" id="MathJax-Span-1510">
               <span class="mo" id="MathJax-Span-1511">
                −
               </span>
               <span class="mi" id="MathJax-Span-1512">
                ∞
               </span>
              </span>
             </span>
             <span class="mtd" id="MathJax-Span-1497">
              <span class="mrow" id="MathJax-Span-1498">
               <span class="msubsup" id="MathJax-Span-1499">
                <span class="mi" id="MathJax-Span-1500">
                 A
                </span>
                <span class="mi" id="MathJax-Span-1501">
                 T
                </span>
               </span>
               <span class="mi" id="MathJax-Span-1502">
                ν
               </span>
               <span class="mo" id="MathJax-Span-1503">
                +
               </span>
               <span class="mi" id="MathJax-Span-1504">
                c
               </span>
               <span class="mo" id="MathJax-Span-1505">
                −
               </span>
               <span class="mi" id="MathJax-Span-1506">
                λ
               </span>
               <span class="mo" id="MathJax-Span-1507">
                =
               </span>
               <span class="mn" id="MathJax-Span-1508">
                0
               </span>
              </span>
             </span>
             <span class="mtd" id="MathJax-Span-1513">
              <span class="mrow" id="MathJax-Span-1514">
               <span class="mtext" id="MathJax-Span-1515">
                otherwise.
               </span>
              </span>
             </span>
            </span>
           </span>
           <span class="mtext" id="MathJax-Span-1516">
           </span>
          </span>
         </span>
        </span>
       </div>
       <p>
        则可以得到对偶问题为：
       </p>
       <div class="MathJax_Display">
        <span class="MathJax" id="MathJax-Element-76-Frame">
         <span class="math" id="MathJax-Span-1517">
          <span class="mrow" id="MathJax-Span-1518">
           <span class="mtable" id="MathJax-Span-1519">
            <span class="mtd" id="MathJax-Span-1520">
             <span class="mrow" id="MathJax-Span-1521">
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1534">
             <span class="mrow" id="MathJax-Span-1535">
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1553">
             <span class="mrow" id="MathJax-Span-1554">
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1522">
             <span class="mrow" id="MathJax-Span-1523">
              <span class="mtext" id="MathJax-Span-1524">
               maximize
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1536">
             <span class="mrow" id="MathJax-Span-1537">
              <span class="mtext" id="MathJax-Span-1538">
               subject to
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1555">
             <span class="mrow" id="MathJax-Span-1556">
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1525">
             <span class="mrow" id="MathJax-Span-1526">
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1539">
             <span class="mrow" id="MathJax-Span-1540">
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1557">
             <span class="mrow" id="MathJax-Span-1558">
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1527">
             <span class="mrow" id="MathJax-Span-1528">
              <span class="mo" id="MathJax-Span-1529">
               −
              </span>
              <span class="msubsup" id="MathJax-Span-1530">
               <span class="mi" id="MathJax-Span-1531">
                b
               </span>
               <span class="mi" id="MathJax-Span-1532">
                T
               </span>
              </span>
              <span class="mi" id="MathJax-Span-1533">
               ν
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1541">
             <span class="mrow" id="MathJax-Span-1542">
              <span class="msubsup" id="MathJax-Span-1543">
               <span class="mi" id="MathJax-Span-1544">
                A
               </span>
               <span class="mi" id="MathJax-Span-1545">
                T
               </span>
              </span>
              <span class="mi" id="MathJax-Span-1546">
               ν
              </span>
              <span class="mo" id="MathJax-Span-1547">
               +
              </span>
              <span class="mi" id="MathJax-Span-1548">
               c
              </span>
              <span class="mo" id="MathJax-Span-1549">
               −
              </span>
              <span class="mi" id="MathJax-Span-1550">
               λ
              </span>
              <span class="mo" id="MathJax-Span-1551">
               =
              </span>
              <span class="mn" id="MathJax-Span-1552">
               0
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1559">
             <span class="mrow" id="MathJax-Span-1560">
              <span class="mi" id="MathJax-Span-1561">
               λ
              </span>
              <span class="mo" id="MathJax-Span-1562">
               ⪰
              </span>
              <span class="mn" id="MathJax-Span-1563">
               0
              </span>
             </span>
            </span>
           </span>
           <span class="mtext" id="MathJax-Span-1564">
           </span>
          </span>
         </span>
        </span>
       </div>
       <p>
        也可以把这个问题写成这样的形式;
       </p>
       <div class="MathJax_Display">
        <span class="MathJax" id="MathJax-Element-77-Frame">
         <span class="math" id="MathJax-Span-1565">
          <span class="mrow" id="MathJax-Span-1566">
           <span class="mtable" id="MathJax-Span-1567">
            <span class="mtd" id="MathJax-Span-1568">
             <span class="mrow" id="MathJax-Span-1569">
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1582">
             <span class="mrow" id="MathJax-Span-1583">
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1570">
             <span class="mrow" id="MathJax-Span-1571">
              <span class="mtext" id="MathJax-Span-1572">
               maximize
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1584">
             <span class="mrow" id="MathJax-Span-1585">
              <span class="mtext" id="MathJax-Span-1586">
               subject to
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1573">
             <span class="mrow" id="MathJax-Span-1574">
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1587">
             <span class="mrow" id="MathJax-Span-1588">
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1575">
             <span class="mrow" id="MathJax-Span-1576">
              <span class="mo" id="MathJax-Span-1577">
               −
              </span>
              <span class="msubsup" id="MathJax-Span-1578">
               <span class="mi" id="MathJax-Span-1579">
                b
               </span>
               <span class="mi" id="MathJax-Span-1580">
                T
               </span>
              </span>
              <span class="mi" id="MathJax-Span-1581">
               ν
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1589">
             <span class="mrow" id="MathJax-Span-1590">
              <span class="msubsup" id="MathJax-Span-1591">
               <span class="mi" id="MathJax-Span-1592">
                A
               </span>
               <span class="mi" id="MathJax-Span-1593">
                T
               </span>
              </span>
              <span class="mi" id="MathJax-Span-1594">
               ν
              </span>
              <span class="mo" id="MathJax-Span-1595">
               +
              </span>
              <span class="mi" id="MathJax-Span-1596">
               c
              </span>
              <span class="mo" id="MathJax-Span-1597">
               ⪰
              </span>
              <span class="mn" id="MathJax-Span-1598">
               0
              </span>
             </span>
            </span>
           </span>
           <span class="mtext" id="MathJax-Span-1599">
           </span>
          </span>
         </span>
        </span>
       </div>
       <h4>
        共轭函数法
        <span id="anchor_JTI1dTUxNzElMjV1OEY2RCUyNXU1MUZEJTI1dTY1NzAlMjV1NkNENTEw">
        </span>
       </h4>
       <p>
        由于线性约束的问题和共轭函数有密切的关系，很多时候我们可以利用共轭函数来求解对偶问题的约束。
       </p>
       <p>
        例如最大化熵问题：
       </p>
       <div class="MathJax_Display">
        <span class="MathJax" id="MathJax-Element-78-Frame">
         <span class="math" id="MathJax-Span-1600">
          <span class="mrow" id="MathJax-Span-1601">
           <span class="mtable" id="MathJax-Span-1602">
            <span class="mtd" id="MathJax-Span-1603">
             <span class="mrow" id="MathJax-Span-1604">
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1635">
             <span class="mrow" id="MathJax-Span-1636">
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1648">
             <span class="mrow" id="MathJax-Span-1649">
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1605">
             <span class="mrow" id="MathJax-Span-1606">
              <span class="mtext" id="MathJax-Span-1607">
               minimize
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1637">
             <span class="mrow" id="MathJax-Span-1638">
              <span class="mtext" id="MathJax-Span-1639">
               subject to
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1650">
             <span class="mrow" id="MathJax-Span-1651">
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1608">
             <span class="mrow" id="MathJax-Span-1609">
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1640">
             <span class="mrow" id="MathJax-Span-1641">
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1652">
             <span class="mrow" id="MathJax-Span-1653">
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1610">
             <span class="mrow" id="MathJax-Span-1611">
              <span class="msubsup" id="MathJax-Span-1612">
               <span class="mi" id="MathJax-Span-1613">
                f
               </span>
               <span class="mn" id="MathJax-Span-1614">
                0
               </span>
              </span>
              <span class="mo" id="MathJax-Span-1615">
               (
              </span>
              <span class="mi" id="MathJax-Span-1616">
               x
              </span>
              <span class="mo" id="MathJax-Span-1617">
               )
              </span>
              <span class="mo" id="MathJax-Span-1618">
               =
              </span>
              <span class="munderover" id="MathJax-Span-1619">
               <span class="mo" id="MathJax-Span-1620">
                ∑
               </span>
               <span class="texatom" id="MathJax-Span-1621">
                <span class="mrow" id="MathJax-Span-1622">
                 <span class="mi" id="MathJax-Span-1623">
                  i
                 </span>
                 <span class="mo" id="MathJax-Span-1624">
                  =
                 </span>
                 <span class="mn" id="MathJax-Span-1625">
                  1
                 </span>
                </span>
               </span>
               <span class="mi" id="MathJax-Span-1626">
                n
               </span>
              </span>
              <span class="msubsup" id="MathJax-Span-1627">
               <span class="mi" id="MathJax-Span-1628">
                x
               </span>
               <span class="mi" id="MathJax-Span-1629">
                i
               </span>
              </span>
              <span class="mi" id="MathJax-Span-1630">
               log
              </span>
              <span class="mo" id="MathJax-Span-1631">
              </span>
              <span class="msubsup" id="MathJax-Span-1632">
               <span class="mi" id="MathJax-Span-1633">
                x
               </span>
               <span class="mi" id="MathJax-Span-1634">
                i
               </span>
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1642">
             <span class="mrow" id="MathJax-Span-1643">
              <span class="mi" id="MathJax-Span-1644">
               A
              </span>
              <span class="mi" id="MathJax-Span-1645">
               x
              </span>
              <span class="mo" id="MathJax-Span-1646">
               ≤
              </span>
              <span class="mi" id="MathJax-Span-1647">
               b
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1654">
             <span class="mrow" id="MathJax-Span-1655">
              <span class="msubsup" id="MathJax-Span-1656">
               <span class="texatom" id="MathJax-Span-1657">
                <span class="mrow" id="MathJax-Span-1658">
                 <span class="mn" id="MathJax-Span-1659">
                  1
                 </span>
                </span>
               </span>
               <span class="mi" id="MathJax-Span-1660">
                T
               </span>
              </span>
              <span class="mi" id="MathJax-Span-1661">
               x
              </span>
              <span class="mo" id="MathJax-Span-1662">
               =
              </span>
              <span class="mn" id="MathJax-Span-1663">
               1
              </span>
             </span>
            </span>
           </span>
           <span class="mtext" id="MathJax-Span-1664">
           </span>
          </span>
         </span>
        </span>
       </div>
       <p>
        我们知道，负熵函数
        <span class="editor-math-element" data-id="JTI1MjRmJTI1Mjh4JTI1MjklMjUzRHglMjU1Q2xvZyUyNTdCeCUyNTdEJTI1MjQ=">
         <span class="MathJax" id="MathJax-Element-79-Frame">
          <span class="math" id="MathJax-Span-1665">
           <span class="mrow" id="MathJax-Span-1666">
            <span class="mi" id="MathJax-Span-1667">
             f
            </span>
            <span class="mo" id="MathJax-Span-1668">
             (
            </span>
            <span class="mi" id="MathJax-Span-1669">
             x
            </span>
            <span class="mo" id="MathJax-Span-1670">
             )
            </span>
            <span class="mo" id="MathJax-Span-1671">
             =
            </span>
            <span class="mi" id="MathJax-Span-1672">
             x
            </span>
            <span class="mi" id="MathJax-Span-1673">
             log
            </span>
            <span class="texatom" id="MathJax-Span-1674">
             <span class="mrow" id="MathJax-Span-1675">
              <span class="mi" id="MathJax-Span-1676">
               x
              </span>
             </span>
            </span>
           </span>
          </span>
         </span>
        </span>
        的共轭函数是
        <span class="editor-math-element" data-id="JTI1MjRmJTI1NUUlMjU1Q2FzdCUyNTI4eSUyNTI5JTI1M0QlMjU1Q21hdGhybSUyNTdCZSUyNTdEJTI1NUUlMjU3QnktMSUyNTdEJTI1MjQ=">
         <span class="MathJax" id="MathJax-Element-80-Frame">
          <span class="math" id="MathJax-Span-1677">
           <span class="mrow" id="MathJax-Span-1678">
            <span class="msubsup" id="MathJax-Span-1679">
             <span class="mi" id="MathJax-Span-1680">
              f
             </span>
             <span class="mo" id="MathJax-Span-1681">
              ∗
             </span>
            </span>
            <span class="mo" id="MathJax-Span-1682">
             (
            </span>
            <span class="mi" id="MathJax-Span-1683">
             y
            </span>
            <span class="mo" id="MathJax-Span-1684">
             )
            </span>
            <span class="mo" id="MathJax-Span-1685">
             =
            </span>
            <span class="msubsup" id="MathJax-Span-1686">
             <span class="texatom" id="MathJax-Span-1687">
              <span class="mrow" id="MathJax-Span-1688">
               <span class="mi" id="MathJax-Span-1689">
                e
               </span>
              </span>
             </span>
             <span class="texatom" id="MathJax-Span-1690">
              <span class="mrow" id="MathJax-Span-1691">
               <span class="mi" id="MathJax-Span-1692">
                y
               </span>
               <span class="mo" id="MathJax-Span-1693">
                −
               </span>
               <span class="mn" id="MathJax-Span-1694">
                1
               </span>
              </span>
             </span>
            </span>
           </span>
          </span>
         </span>
        </span>
        ，定义域是
        <span class="editor-math-element" data-id="JTI1MjQlMjU1Q2JmJTI1MjBSJTI1MjQ=">
         <span class="MathJax" id="MathJax-Element-81-Frame">
          <span class="math" id="MathJax-Span-1695">
           <span class="mrow" id="MathJax-Span-1696">
            <span class="mi" id="MathJax-Span-1697">
             R
            </span>
           </span>
          </span>
         </span>
        </span>
        。可以看出上面
        <span class="editor-math-element" data-id="JTI1MjRmXzAlMjUyNA==">
         <span class="MathJax" id="MathJax-Element-82-Frame">
          <span class="math" id="MathJax-Span-1698">
           <span class="mrow" id="MathJax-Span-1699">
            <span class="msubsup" id="MathJax-Span-1700">
             <span class="mi" id="MathJax-Span-1701">
              f
             </span>
             <span class="mn" id="MathJax-Span-1702">
              0
             </span>
            </span>
           </span>
          </span>
         </span>
        </span>
        的共轭函数是：
       </p>
       <div class="MathJax_Display">
        <span class="MathJax" id="MathJax-Element-83-Frame">
         <span class="math" id="MathJax-Span-1703">
          <span class="mrow" id="MathJax-Span-1704">
           <span class="msubsup" id="MathJax-Span-1705">
            <span class="mi" id="MathJax-Span-1706">
             f
            </span>
            <span class="mo" id="MathJax-Span-1707">
             ∗
            </span>
            <span class="mn" id="MathJax-Span-1708">
             0
            </span>
           </span>
           <span class="mo" id="MathJax-Span-1709">
            (
           </span>
           <span class="mi" id="MathJax-Span-1710">
            y
           </span>
           <span class="mo" id="MathJax-Span-1711">
            )
           </span>
           <span class="mo" id="MathJax-Span-1712">
            =
           </span>
           <span class="munderover" id="MathJax-Span-1713">
            <span class="mo" id="MathJax-Span-1714">
             ∑
            </span>
            <span class="texatom" id="MathJax-Span-1715">
             <span class="mrow" id="MathJax-Span-1716">
              <span class="mi" id="MathJax-Span-1717">
               i
              </span>
              <span class="mo" id="MathJax-Span-1718">
               =
              </span>
              <span class="mn" id="MathJax-Span-1719">
               1
              </span>
             </span>
            </span>
            <span class="mi" id="MathJax-Span-1720">
             n
            </span>
           </span>
           <span class="msubsup" id="MathJax-Span-1721">
            <span class="texatom" id="MathJax-Span-1722">
             <span class="mrow" id="MathJax-Span-1723">
              <span class="mi" id="MathJax-Span-1724">
               e
              </span>
             </span>
            </span>
            <span class="texatom" id="MathJax-Span-1725">
             <span class="mrow" id="MathJax-Span-1726">
              <span class="msubsup" id="MathJax-Span-1727">
               <span class="mi" id="MathJax-Span-1728">
                y
               </span>
               <span class="mi" id="MathJax-Span-1729">
                i
               </span>
              </span>
              <span class="mo" id="MathJax-Span-1730">
               −
              </span>
              <span class="mn" id="MathJax-Span-1731">
               1
              </span>
             </span>
            </span>
           </span>
           <span class="mtext" id="MathJax-Span-1732">
           </span>
          </span>
         </span>
        </span>
       </div>
       <p>
        定义域是
        <span class="editor-math-element" data-id="JTI1MjQlMjU3QiUyNTVDYmYlMjUyMFIlMjU3RCUyNTVFbiUyNTI0">
         <span class="MathJax" id="MathJax-Element-84-Frame">
          <span class="math" id="MathJax-Span-1733">
           <span class="mrow" id="MathJax-Span-1734">
            <span class="msubsup" id="MathJax-Span-1735">
             <span class="texatom" id="MathJax-Span-1736">
              <span class="mrow" id="MathJax-Span-1737">
               <span class="mi" id="MathJax-Span-1738">
                R
               </span>
              </span>
             </span>
             <span class="mi" id="MathJax-Span-1739">
              n
             </span>
            </span>
           </span>
          </span>
         </span>
        </span>
        。
        <br/>
        由线性约束问题的对偶函数与共轭函数的关系可得：
       </p>
       <div class="MathJax_Display">
        <span class="MathJax" id="MathJax-Element-85-Frame">
         <span class="math" id="MathJax-Span-1740">
          <span class="mrow" id="MathJax-Span-1741">
           <span class="mi" id="MathJax-Span-1742">
            g
           </span>
           <span class="mo" id="MathJax-Span-1743">
            (
           </span>
           <span class="texatom" id="MathJax-Span-1744">
            <span class="mrow" id="MathJax-Span-1745">
             <span class="mi" id="MathJax-Span-1746">
              λ
             </span>
             <span class="mo" id="MathJax-Span-1747">
              ,
             </span>
             <span class="mi" id="MathJax-Span-1748">
              ν
             </span>
            </span>
           </span>
           <span class="mo" id="MathJax-Span-1749">
            =
           </span>
           <span class="mo" id="MathJax-Span-1750">
            −
           </span>
           <span class="msubsup" id="MathJax-Span-1751">
            <span class="mi" id="MathJax-Span-1752">
             b
            </span>
            <span class="mi" id="MathJax-Span-1753">
             T
            </span>
           </span>
           <span class="mi" id="MathJax-Span-1754">
            λ
           </span>
           <span class="mo" id="MathJax-Span-1755">
            −
           </span>
           <span class="mi" id="MathJax-Span-1756">
            ν
           </span>
           <span class="mo" id="MathJax-Span-1757">
            −
           </span>
           <span class="munderover" id="MathJax-Span-1758">
            <span class="mo" id="MathJax-Span-1759">
             ∑
            </span>
            <span class="texatom" id="MathJax-Span-1760">
             <span class="mrow" id="MathJax-Span-1761">
              <span class="mi" id="MathJax-Span-1762">
               i
              </span>
              <span class="mo" id="MathJax-Span-1763">
               =
              </span>
              <span class="mn" id="MathJax-Span-1764">
               1
              </span>
             </span>
            </span>
            <span class="mi" id="MathJax-Span-1765">
             n
            </span>
           </span>
           <span class="msubsup" id="MathJax-Span-1766">
            <span class="texatom" id="MathJax-Span-1767">
             <span class="mrow" id="MathJax-Span-1768">
              <span class="mi" id="MathJax-Span-1769">
               e
              </span>
             </span>
            </span>
            <span class="texatom" id="MathJax-Span-1770">
             <span class="mrow" id="MathJax-Span-1771">
              <span class="mo" id="MathJax-Span-1772">
               −
              </span>
              <span class="msubsup" id="MathJax-Span-1773">
               <span class="mi" id="MathJax-Span-1774">
                a
               </span>
               <span class="mi" id="MathJax-Span-1775">
                T
               </span>
               <span class="mi" id="MathJax-Span-1776">
                i
               </span>
              </span>
              <span class="mi" id="MathJax-Span-1777">
               λ
              </span>
              <span class="mo" id="MathJax-Span-1778">
               −
              </span>
              <span class="mi" id="MathJax-Span-1779">
               ν
              </span>
              <span class="mo" id="MathJax-Span-1780">
               −
              </span>
              <span class="mn" id="MathJax-Span-1781">
               1
              </span>
             </span>
            </span>
           </span>
           <span class="mo" id="MathJax-Span-1782">
            =
           </span>
           <span class="mo" id="MathJax-Span-1783">
            −
           </span>
           <span class="msubsup" id="MathJax-Span-1784">
            <span class="mi" id="MathJax-Span-1785">
             b
            </span>
            <span class="mi" id="MathJax-Span-1786">
             T
            </span>
           </span>
           <span class="mi" id="MathJax-Span-1787">
            λ
           </span>
           <span class="mo" id="MathJax-Span-1788">
            −
           </span>
           <span class="mi" id="MathJax-Span-1789">
            ν
           </span>
           <span class="mo" id="MathJax-Span-1790">
            −
           </span>
           <span class="msubsup" id="MathJax-Span-1791">
            <span class="texatom" id="MathJax-Span-1792">
             <span class="mrow" id="MathJax-Span-1793">
              <span class="mi" id="MathJax-Span-1794">
               e
              </span>
             </span>
            </span>
            <span class="texatom" id="MathJax-Span-1795">
             <span class="mrow" id="MathJax-Span-1796">
              <span class="mo" id="MathJax-Span-1797">
               −
              </span>
              <span class="mi" id="MathJax-Span-1798">
               ν
              </span>
              <span class="mo" id="MathJax-Span-1799">
               −
              </span>
              <span class="mn" id="MathJax-Span-1800">
               1
              </span>
             </span>
            </span>
           </span>
           <span class="munderover" id="MathJax-Span-1801">
            <span class="mo" id="MathJax-Span-1802">
             ∑
            </span>
            <span class="texatom" id="MathJax-Span-1803">
             <span class="mrow" id="MathJax-Span-1804">
              <span class="mi" id="MathJax-Span-1805">
               i
              </span>
              <span class="mo" id="MathJax-Span-1806">
               =
              </span>
              <span class="mn" id="MathJax-Span-1807">
               1
              </span>
             </span>
            </span>
            <span class="mi" id="MathJax-Span-1808">
             n
            </span>
           </span>
           <span class="msubsup" id="MathJax-Span-1809">
            <span class="texatom" id="MathJax-Span-1810">
             <span class="mrow" id="MathJax-Span-1811">
              <span class="mi" id="MathJax-Span-1812">
               e
              </span>
             </span>
            </span>
            <span class="texatom" id="MathJax-Span-1813">
             <span class="mrow" id="MathJax-Span-1814">
              <span class="mo" id="MathJax-Span-1815">
               −
              </span>
              <span class="msubsup" id="MathJax-Span-1816">
               <span class="mi" id="MathJax-Span-1817">
                a
               </span>
               <span class="mi" id="MathJax-Span-1818">
                T
               </span>
               <span class="mi" id="MathJax-Span-1819">
                i
               </span>
              </span>
              <span class="mi" id="MathJax-Span-1820">
               λ
              </span>
             </span>
            </span>
           </span>
           <span class="mtext" id="MathJax-Span-1821">
           </span>
          </span>
         </span>
        </span>
       </div>
       <p>
        这里
        <span class="editor-math-element" data-id="JTI1MjRhX2klMjUyNA==">
         <span class="MathJax" id="MathJax-Element-86-Frame">
          <span class="math" id="MathJax-Span-1822">
           <span class="mrow" id="MathJax-Span-1823">
            <span class="msubsup" id="MathJax-Span-1824">
             <span class="mi" id="MathJax-Span-1825">
              a
             </span>
             <span class="mi" id="MathJax-Span-1826">
              i
             </span>
            </span>
           </span>
          </span>
         </span>
        </span>
        是
        <span class="editor-math-element" data-id="JTI1MjQlMjU1Q2JmJTI1MjBBJTI1MjQ=">
         <span class="MathJax" id="MathJax-Element-87-Frame">
          <span class="math" id="MathJax-Span-1827">
           <span class="mrow" id="MathJax-Span-1828">
            <span class="mi" id="MathJax-Span-1829">
             A
            </span>
           </span>
          </span>
         </span>
        </span>
        矩阵的第i列。
       </p>
       <p>
        所以，这个问题也转化为了一个无约束的优化问题：
       </p>
       <div class="MathJax_Display">
        <span class="MathJax" id="MathJax-Element-88-Frame">
         <span class="math" id="MathJax-Span-1830">
          <span class="mrow" id="MathJax-Span-1831">
           <span class="mtext" id="MathJax-Span-1832">
            maximize
           </span>
           <span class="mspace" id="MathJax-Span-1833">
           </span>
           <span class="mo" id="MathJax-Span-1834">
            −
           </span>
           <span class="msubsup" id="MathJax-Span-1835">
            <span class="mi" id="MathJax-Span-1836">
             b
            </span>
            <span class="mi" id="MathJax-Span-1837">
             T
            </span>
           </span>
           <span class="mi" id="MathJax-Span-1838">
            λ
           </span>
           <span class="mo" id="MathJax-Span-1839">
            −
           </span>
           <span class="mi" id="MathJax-Span-1840">
            ν
           </span>
           <span class="mo" id="MathJax-Span-1841">
            −
           </span>
           <span class="msubsup" id="MathJax-Span-1842">
            <span class="texatom" id="MathJax-Span-1843">
             <span class="mrow" id="MathJax-Span-1844">
              <span class="mi" id="MathJax-Span-1845">
               e
              </span>
             </span>
            </span>
            <span class="texatom" id="MathJax-Span-1846">
             <span class="mrow" id="MathJax-Span-1847">
              <span class="mo" id="MathJax-Span-1848">
               −
              </span>
              <span class="mi" id="MathJax-Span-1849">
               ν
              </span>
              <span class="mo" id="MathJax-Span-1850">
               −
              </span>
              <span class="mn" id="MathJax-Span-1851">
               1
              </span>
             </span>
            </span>
           </span>
           <span class="munderover" id="MathJax-Span-1852">
            <span class="mo" id="MathJax-Span-1853">
             ∑
            </span>
            <span class="texatom" id="MathJax-Span-1854">
             <span class="mrow" id="MathJax-Span-1855">
              <span class="mi" id="MathJax-Span-1856">
               i
              </span>
              <span class="mo" id="MathJax-Span-1857">
               =
              </span>
              <span class="mn" id="MathJax-Span-1858">
               1
              </span>
             </span>
            </span>
            <span class="mi" id="MathJax-Span-1859">
             n
            </span>
           </span>
           <span class="msubsup" id="MathJax-Span-1860">
            <span class="texatom" id="MathJax-Span-1861">
             <span class="mrow" id="MathJax-Span-1862">
              <span class="mi" id="MathJax-Span-1863">
               e
              </span>
             </span>
            </span>
            <span class="texatom" id="MathJax-Span-1864">
             <span class="mrow" id="MathJax-Span-1865">
              <span class="mo" id="MathJax-Span-1866">
               −
              </span>
              <span class="msubsup" id="MathJax-Span-1867">
               <span class="mi" id="MathJax-Span-1868">
                a
               </span>
               <span class="mi" id="MathJax-Span-1869">
                T
               </span>
               <span class="mi" id="MathJax-Span-1870">
                i
               </span>
              </span>
              <span class="mi" id="MathJax-Span-1871">
               λ
              </span>
             </span>
            </span>
           </span>
           <span class="mtext" id="MathJax-Span-1872">
           </span>
          </span>
         </span>
        </span>
       </div>
       <h2>
        弱对偶
        <span id="anchor_JTI1dTVGMzElMjV1NUJGOSUyNXU1MDc2MTE=">
        </span>
       </h2>
       <p>
        如果我们把拉格朗日对偶问题的最优值记为
        <span class="editor-math-element" data-id="JTI1MjRkJTI1NUUlMjU1Q3N0YXIlMjUyNA==">
         <span class="MathJax" id="MathJax-Element-89-Frame">
          <span class="math" id="MathJax-Span-1873">
           <span class="mrow" id="MathJax-Span-1874">
            <span class="msubsup" id="MathJax-Span-1875">
             <span class="mi" id="MathJax-Span-1876">
              d
             </span>
             <span class="mo" id="MathJax-Span-1877">
              ⋆
             </span>
            </span>
           </span>
          </span>
         </span>
        </span>
        。相对于原始问题的最优值
        <span class="editor-math-element" data-id="JTI1MjRwJTI1NUUlMjU1Q3N0YXIlMjUyNA==">
         <span class="MathJax" id="MathJax-Element-90-Frame">
          <span class="math" id="MathJax-Span-1878">
           <span class="mrow" id="MathJax-Span-1879">
            <span class="msubsup" id="MathJax-Span-1880">
             <span class="mi" id="MathJax-Span-1881">
              p
             </span>
             <span class="mo" id="MathJax-Span-1882">
              ⋆
             </span>
            </span>
           </span>
          </span>
         </span>
        </span>
        ，我们有如下关系：
       </p>
       <div class="MathJax_Display">
        <span class="MathJax" id="MathJax-Element-91-Frame">
         <span class="math" id="MathJax-Span-1883">
          <span class="mrow" id="MathJax-Span-1884">
           <span class="msubsup" id="MathJax-Span-1885">
            <span class="mi" id="MathJax-Span-1886">
             d
            </span>
            <span class="mo" id="MathJax-Span-1887">
             ⋆
            </span>
           </span>
           <span class="mo" id="MathJax-Span-1888">
            ≤
           </span>
           <span class="msubsup" id="MathJax-Span-1889">
            <span class="mi" id="MathJax-Span-1890">
             p
            </span>
            <span class="mo" id="MathJax-Span-1891">
             ⋆
            </span>
           </span>
           <span class="mtext" id="MathJax-Span-1892">
           </span>
          </span>
         </span>
        </span>
       </div>
       <p>
        这个关系就被称为弱对偶关系。
       </p>
       <p>
        弱对偶关系即使在
        <span class="editor-math-element" data-id="JTI1MjRkJTI1NUUlMjU1Q3N0YXIlMjUyNA==">
         <span class="MathJax" id="MathJax-Element-92-Frame">
          <span class="math" id="MathJax-Span-1893">
           <span class="mrow" id="MathJax-Span-1894">
            <span class="msubsup" id="MathJax-Span-1895">
             <span class="mi" id="MathJax-Span-1896">
              d
             </span>
             <span class="mo" id="MathJax-Span-1897">
              ⋆
             </span>
            </span>
           </span>
          </span>
         </span>
        </span>
        或
        <span class="editor-math-element" data-id="JTI1MjRwJTI1NUUlMjU1Q3N0YXIlMjUyNA==">
         <span class="MathJax" id="MathJax-Element-93-Frame">
          <span class="math" id="MathJax-Span-1898">
           <span class="mrow" id="MathJax-Span-1899">
            <span class="msubsup" id="MathJax-Span-1900">
             <span class="mi" id="MathJax-Span-1901">
              p
             </span>
             <span class="mo" id="MathJax-Span-1902">
              ⋆
             </span>
            </span>
           </span>
          </span>
         </span>
        </span>
        为无穷的时候也是成立的。如果原始问题是无界的，即
        <span class="editor-math-element" data-id="JTI1MjRwJTI1NUUlMjU1Q3N0YXIlMjUyMCUyNTNEJTI1MjAtJTI1NUNpbmZ0eSUyNTI0">
         <span class="MathJax" id="MathJax-Element-94-Frame">
          <span class="math" id="MathJax-Span-1903">
           <span class="mrow" id="MathJax-Span-1904">
            <span class="msubsup" id="MathJax-Span-1905">
             <span class="mi" id="MathJax-Span-1906">
              p
             </span>
             <span class="mo" id="MathJax-Span-1907">
              ⋆
             </span>
            </span>
            <span class="mo" id="MathJax-Span-1908">
             =
            </span>
            <span class="mo" id="MathJax-Span-1909">
             −
            </span>
            <span class="mi" id="MathJax-Span-1910">
             ∞
            </span>
           </span>
          </span>
         </span>
        </span>
        ,则
        <span class="editor-math-element" data-id="JTI1MjRkJTI1NUUlMjU1Q3N0YXIlMjUzRC0lMjU1Q2luZnR5JTI1MjQ=">
         <span class="MathJax" id="MathJax-Element-95-Frame">
          <span class="math" id="MathJax-Span-1911">
           <span class="mrow" id="MathJax-Span-1912">
            <span class="msubsup" id="MathJax-Span-1913">
             <span class="mi" id="MathJax-Span-1914">
              d
             </span>
             <span class="mo" id="MathJax-Span-1915">
              ⋆
             </span>
            </span>
            <span class="mo" id="MathJax-Span-1916">
             =
            </span>
            <span class="mo" id="MathJax-Span-1917">
             −
            </span>
            <span class="mi" id="MathJax-Span-1918">
             ∞
            </span>
           </span>
          </span>
         </span>
        </span>
        ,即拉格朗日对偶问题是不可行的。相应的，如果对偶问题无界，即
        <span class="editor-math-element" data-id="JTI1MjRkJTI1NUUlMjU1Q3N0YXIlMjUzRCUyNTVDaW5mdHklMjUyNA==">
         <span class="MathJax" id="MathJax-Element-96-Frame">
          <span class="math" id="MathJax-Span-1919">
           <span class="mrow" id="MathJax-Span-1920">
            <span class="msubsup" id="MathJax-Span-1921">
             <span class="mi" id="MathJax-Span-1922">
              d
             </span>
             <span class="mo" id="MathJax-Span-1923">
              ⋆
             </span>
            </span>
            <span class="mo" id="MathJax-Span-1924">
             =
            </span>
            <span class="mi" id="MathJax-Span-1925">
             ∞
            </span>
           </span>
          </span>
         </span>
        </span>
        ，则
        <span class="editor-math-element" data-id="JTI1MjRwJTI1NUUlMjU1Q3N0YXIlMjUyMCUyNTNEJTI1MjAlMjU1Q2luZnR5JTI1MjQ=">
         <span class="MathJax" id="MathJax-Element-97-Frame">
          <span class="math" id="MathJax-Span-1926">
           <span class="mrow" id="MathJax-Span-1927">
            <span class="msubsup" id="MathJax-Span-1928">
             <span class="mi" id="MathJax-Span-1929">
              p
             </span>
             <span class="mo" id="MathJax-Span-1930">
              ⋆
             </span>
            </span>
            <span class="mo" id="MathJax-Span-1931">
             =
            </span>
            <span class="mi" id="MathJax-Span-1932">
             ∞
            </span>
           </span>
          </span>
         </span>
        </span>
        ，即原始问题是不可行的。
       </p>
       <p>
        我们把原始问题和对偶问题最优值之间的差值
        <span class="editor-math-element" data-id="JTI1MjRwJTI1NUUlMjU1Q3N0YXIlMjUyMC0lMjUyMGQlMjU1RSUyNTVDc3RhciUyNTI0">
         <span class="MathJax" id="MathJax-Element-98-Frame">
          <span class="math" id="MathJax-Span-1933">
           <span class="mrow" id="MathJax-Span-1934">
            <span class="msubsup" id="MathJax-Span-1935">
             <span class="mi" id="MathJax-Span-1936">
              p
             </span>
             <span class="mo" id="MathJax-Span-1937">
              ⋆
             </span>
            </span>
            <span class="mo" id="MathJax-Span-1938">
             −
            </span>
            <span class="msubsup" id="MathJax-Span-1939">
             <span class="mi" id="MathJax-Span-1940">
              d
             </span>
             <span class="mo" id="MathJax-Span-1941">
              ⋆
             </span>
            </span>
           </span>
          </span>
         </span>
        </span>
        称为最有对偶间隙（optimal duality gap）。显然，这个值总是正的。
       </p>
       <h2>
        强对偶
        <span id="anchor_JTI1dTVGM0ElMjV1NUJGOSUyNXU1MDc2MTI=">
        </span>
       </h2>
       <p>
        如果原始问题和对偶问题的最优值相等，即：
       </p>
       <div class="MathJax_Display">
        <span class="MathJax" id="MathJax-Element-99-Frame">
         <span class="math" id="MathJax-Span-1942">
          <span class="mrow" id="MathJax-Span-1943">
           <span class="msubsup" id="MathJax-Span-1944">
            <span class="mi" id="MathJax-Span-1945">
             d
            </span>
            <span class="mo" id="MathJax-Span-1946">
             ⋆
            </span>
           </span>
           <span class="mo" id="MathJax-Span-1947">
            =
           </span>
           <span class="msubsup" id="MathJax-Span-1948">
            <span class="mi" id="MathJax-Span-1949">
             p
            </span>
            <span class="mo" id="MathJax-Span-1950">
             ⋆
            </span>
           </span>
           <span class="mtext" id="MathJax-Span-1951">
           </span>
          </span>
         </span>
        </span>
       </div>
       <p>
        我们就称这种关系为强对偶关系。在这种情况下，我们仅仅从对偶问题中就可以知道原始问题的最优解。
       </p>
       <p>
        通常情况下强对偶关系并不成立。但是如果原始问题是凸的，即对于这样的形式：
       </p>
       <div class="MathJax_Display">
        <span class="MathJax" id="MathJax-Element-100-Frame">
         <span class="math" id="MathJax-Span-1952">
          <span class="mrow" id="MathJax-Span-1953">
           <span class="mtable" id="MathJax-Span-1954">
            <span class="mtd" id="MathJax-Span-1955">
             <span class="mrow" id="MathJax-Span-1956">
              <span class="mtext" id="MathJax-Span-1957">
               minimize
              </span>
              <span class="mspace" id="MathJax-Span-1958">
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1969">
             <span class="mrow" id="MathJax-Span-1970">
              <span class="mtext" id="MathJax-Span-1971">
               subject to
              </span>
              <span class="mspace" id="MathJax-Span-1972">
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1994">
             <span class="mrow" id="MathJax-Span-1995">
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1959">
             <span class="mrow" id="MathJax-Span-1960">
              <span class="msubsup" id="MathJax-Span-1961">
               <span class="mi" id="MathJax-Span-1962">
                f
               </span>
               <span class="mn" id="MathJax-Span-1963">
                0
               </span>
              </span>
              <span class="mo" id="MathJax-Span-1964">
               (
              </span>
              <span class="texatom" id="MathJax-Span-1965">
               <span class="mrow" id="MathJax-Span-1966">
                <span class="mi" id="MathJax-Span-1967">
                 x
                </span>
               </span>
              </span>
              <span class="mo" id="MathJax-Span-1968">
               )
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1973">
             <span class="mrow" id="MathJax-Span-1974">
              <span class="msubsup" id="MathJax-Span-1975">
               <span class="mi" id="MathJax-Span-1976">
                f
               </span>
               <span class="mi" id="MathJax-Span-1977">
                i
               </span>
              </span>
              <span class="mo" id="MathJax-Span-1978">
               (
              </span>
              <span class="texatom" id="MathJax-Span-1979">
               <span class="mrow" id="MathJax-Span-1980">
                <span class="mi" id="MathJax-Span-1981">
                 x
                </span>
               </span>
              </span>
              <span class="mo" id="MathJax-Span-1982">
               )
              </span>
              <span class="mo" id="MathJax-Span-1983">
               ≤
              </span>
              <span class="mn" id="MathJax-Span-1984">
               0
              </span>
              <span class="mo" id="MathJax-Span-1985">
               ,
              </span>
              <span class="mspace" id="MathJax-Span-1986">
              </span>
              <span class="mi" id="MathJax-Span-1987">
               i
              </span>
              <span class="mo" id="MathJax-Span-1988">
               =
              </span>
              <span class="mn" id="MathJax-Span-1989">
               1
              </span>
              <span class="mo" id="MathJax-Span-1990">
               ,
              </span>
              <span class="mo" id="MathJax-Span-1991">
               …
              </span>
              <span class="mo" id="MathJax-Span-1992">
               ,
              </span>
              <span class="mi" id="MathJax-Span-1993">
               m
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-1996">
             <span class="mrow" id="MathJax-Span-1997">
              <span class="mi" id="MathJax-Span-1998">
               A
              </span>
              <span class="mi" id="MathJax-Span-1999">
               x
              </span>
              <span class="mo" id="MathJax-Span-2000">
               =
              </span>
              <span class="mi" id="MathJax-Span-2001">
               b
              </span>
             </span>
            </span>
           </span>
           <span class="mtext" id="MathJax-Span-2002">
           </span>
          </span>
         </span>
        </span>
       </div>
       <p>
        有
        <span class="editor-math-element" data-id="JTI1MjRmXzAlMjUyQyUyNTVDZG90cyUyNTJDZl9tJTI1MjQ=">
         <span class="MathJax" id="MathJax-Element-101-Frame">
          <span class="math" id="MathJax-Span-2003">
           <span class="mrow" id="MathJax-Span-2004">
            <span class="msubsup" id="MathJax-Span-2005">
             <span class="mi" id="MathJax-Span-2006">
              f
             </span>
             <span class="mn" id="MathJax-Span-2007">
              0
             </span>
            </span>
            <span class="mo" id="MathJax-Span-2008">
             ,
            </span>
            <span class="mo" id="MathJax-Span-2009">
             …
            </span>
            <span class="mo" id="MathJax-Span-2010">
             ,
            </span>
            <span class="msubsup" id="MathJax-Span-2011">
             <span class="mi" id="MathJax-Span-2012">
              f
             </span>
             <span class="mi" id="MathJax-Span-2013">
              m
             </span>
            </span>
           </span>
          </span>
         </span>
        </span>
        都是凸的，那么我们通常能够得到强对偶关系。但是这种情况下也存在两个问题都无可行解的情况，此时强对偶关系不成立。为了确保我们一定能得到强对偶关系，除了原始问题是凸的之外，我们还需要更多的限制条件。
        <br/>
        一种经常被用到的条件是
       </p>
       <ul>
        <li>
         <strong>
          Slator条件
         </strong>
         ：存在一点
         <span class="editor-math-element" data-id="JTI1MjR4JTI1MjAlMjU1Q2luJTI1MjAlMjU1Q3RleHQlMjU3QnJlbGludCUyNTIwJTI1N0QlMjUyMEQlMjUyNA==">
          <span class="MathJax" id="MathJax-Element-102-Frame">
           <span class="math" id="MathJax-Span-2014">
            <span class="mrow" id="MathJax-Span-2015">
             <span class="mi" id="MathJax-Span-2016">
              x
             </span>
             <span class="mo" id="MathJax-Span-2017">
              ∈
             </span>
             <span class="mtext" id="MathJax-Span-2018">
              relint
             </span>
             <span class="mi" id="MathJax-Span-2019">
              D
             </span>
            </span>
           </span>
          </span>
         </span>
         （D的相对内点集）满足
         <div class="MathJax_Display">
          <span class="MathJax" id="MathJax-Element-103-Frame">
           <span class="math" id="MathJax-Span-2020">
            <span class="mrow" id="MathJax-Span-2021">
             <span class="msubsup" id="MathJax-Span-2022">
              <span class="mi" id="MathJax-Span-2023">
               f
              </span>
              <span class="mi" id="MathJax-Span-2024">
               i
              </span>
             </span>
             <span class="mo" id="MathJax-Span-2025">
              (
             </span>
             <span class="mi" id="MathJax-Span-2026">
              x
             </span>
             <span class="mo" id="MathJax-Span-2027">
              )
             </span>
             <span class="mo" id="MathJax-Span-2028">
              &lt;
             </span>
             <span class="mn" id="MathJax-Span-2029">
              0
             </span>
             <span class="mo" id="MathJax-Span-2030">
              ,
             </span>
             <span class="mspace" id="MathJax-Span-2031">
             </span>
             <span class="mi" id="MathJax-Span-2032">
              i
             </span>
             <span class="mo" id="MathJax-Span-2033">
              =
             </span>
             <span class="mn" id="MathJax-Span-2034">
              1
             </span>
             <span class="mo" id="MathJax-Span-2035">
              ,
             </span>
             <span class="mo" id="MathJax-Span-2036">
              …
             </span>
             <span class="mo" id="MathJax-Span-2037">
              ,
             </span>
             <span class="mi" id="MathJax-Span-2038">
              m
             </span>
             <span class="mo" id="MathJax-Span-2039">
              ,
             </span>
             <span class="mspace" id="MathJax-Span-2040">
             </span>
             <span class="mi" id="MathJax-Span-2041">
              A
             </span>
             <span class="mi" id="MathJax-Span-2042">
              x
             </span>
             <span class="mo" id="MathJax-Span-2043">
              =
             </span>
             <span class="mi" id="MathJax-Span-2044">
              b
             </span>
            </span>
           </span>
          </span>
         </div>
         <p>
          这样的点也可称之为 严格可行 的。
         </p>
        </li>
       </ul>
       <p>
        可以证明，如果原问题是凸的，并且Slator条件成立的情况下，强对偶条件一定成立。
       </p>
       <p>
        如果有一些不等约束是仿射的，Slator条件还可以被弱化。假设前
        <span class="editor-math-element" data-id="JTI1MjRrJTI1MjQ=">
         <span class="MathJax" id="MathJax-Element-104-Frame">
          <span class="math" id="MathJax-Span-2045">
           <span class="mrow" id="MathJax-Span-2046">
            <span class="mi" id="MathJax-Span-2047">
             k
            </span>
           </span>
          </span>
         </span>
        </span>
        个不等约束是仿射的，则Slator条件可以被转化为：
       </p>
       <ul>
        <li>
         <strong>
          Refined Slater’s 条件
         </strong>
         ：存在一点
         <span class="editor-math-element" data-id="JTI1MjR4JTI1MjAlMjU1Q2luJTI1MjAlMjU1Q3RleHQlMjU3QnJlbGludCUyNTIwJTI1N0QlMjUyMEQlMjUyNA==">
          <span class="MathJax" id="MathJax-Element-105-Frame">
           <span class="math" id="MathJax-Span-2048">
            <span class="mrow" id="MathJax-Span-2049">
             <span class="mi" id="MathJax-Span-2050">
              x
             </span>
             <span class="mo" id="MathJax-Span-2051">
              ∈
             </span>
             <span class="mtext" id="MathJax-Span-2052">
              relint
             </span>
             <span class="mi" id="MathJax-Span-2053">
              D
             </span>
            </span>
           </span>
          </span>
         </span>
         （D的相对内点集）满足
         <div class="MathJax_Display">
          <span class="MathJax" id="MathJax-Element-106-Frame">
           <span class="math" id="MathJax-Span-2054">
            <span class="mrow" id="MathJax-Span-2055">
             <span class="msubsup" id="MathJax-Span-2056">
              <span class="mi" id="MathJax-Span-2057">
               f
              </span>
              <span class="mi" id="MathJax-Span-2058">
               i
              </span>
             </span>
             <span class="mo" id="MathJax-Span-2059">
              (
             </span>
             <span class="mi" id="MathJax-Span-2060">
              x
             </span>
             <span class="mo" id="MathJax-Span-2061">
              )
             </span>
             <span class="mo" id="MathJax-Span-2062">
              ≤
             </span>
             <span class="mn" id="MathJax-Span-2063">
              0
             </span>
             <span class="mo" id="MathJax-Span-2064">
              ,
             </span>
             <span class="mspace" id="MathJax-Span-2065">
             </span>
             <span class="mi" id="MathJax-Span-2066">
              i
             </span>
             <span class="mo" id="MathJax-Span-2067">
              =
             </span>
             <span class="mn" id="MathJax-Span-2068">
              1
             </span>
             <span class="mo" id="MathJax-Span-2069">
              ,
             </span>
             <span class="mo" id="MathJax-Span-2070">
              …
             </span>
             <span class="mo" id="MathJax-Span-2071">
              ,
             </span>
             <span class="mi" id="MathJax-Span-2072">
              k
             </span>
             <span class="mo" id="MathJax-Span-2073">
              ,
             </span>
             <span class="mspace" id="MathJax-Span-2074">
             </span>
             <span class="msubsup" id="MathJax-Span-2075">
              <span class="mi" id="MathJax-Span-2076">
               f
              </span>
              <span class="mi" id="MathJax-Span-2077">
               i
              </span>
             </span>
             <span class="mo" id="MathJax-Span-2078">
              (
             </span>
             <span class="mi" id="MathJax-Span-2079">
              x
             </span>
             <span class="mo" id="MathJax-Span-2080">
              )
             </span>
             <span class="mo" id="MathJax-Span-2081">
              &lt;
             </span>
             <span class="mn" id="MathJax-Span-2082">
              0
             </span>
             <span class="mo" id="MathJax-Span-2083">
              ,
             </span>
             <span class="mspace" id="MathJax-Span-2084">
             </span>
             <span class="mi" id="MathJax-Span-2085">
              i
             </span>
             <span class="mo" id="MathJax-Span-2086">
              =
             </span>
             <span class="mi" id="MathJax-Span-2087">
              k
             </span>
             <span class="mo" id="MathJax-Span-2088">
              +
             </span>
             <span class="mn" id="MathJax-Span-2089">
              1
             </span>
             <span class="mo" id="MathJax-Span-2090">
              ,
             </span>
             <span class="mo" id="MathJax-Span-2091">
              …
             </span>
             <span class="mo" id="MathJax-Span-2092">
              ,
             </span>
             <span class="mi" id="MathJax-Span-2093">
              m
             </span>
             <span class="mspace" id="MathJax-Span-2094">
             </span>
             <span class="mi" id="MathJax-Span-2095">
              A
             </span>
             <span class="mi" id="MathJax-Span-2096">
              x
             </span>
             <span class="mo" id="MathJax-Span-2097">
              =
             </span>
             <span class="mi" id="MathJax-Span-2098">
              b
             </span>
            </span>
           </span>
          </span>
         </div>
        </li>
       </ul>
       <p>
        即仿射的不等约束不必取严格小于。
       </p>
       <p>
        此外，满足 Slater’s 条件（或 Refined Slater’s 条件） 不仅意味着（凸优化问题）强对偶性的成立，而且也表示当
        <span class="editor-math-element" data-id="JTI1MjRkJTI1NUUlMjU1Q3N0YXIlMjUyMCUyNTIwJTI1M0UlMjUyMC0lMjU1Q2luZnR5JTI1MjQ=">
         <span class="MathJax" id="MathJax-Element-107-Frame">
          <span class="math" id="MathJax-Span-2099">
           <span class="mrow" id="MathJax-Span-2100">
            <span class="msubsup" id="MathJax-Span-2101">
             <span class="mi" id="MathJax-Span-2102">
              d
             </span>
             <span class="mo" id="MathJax-Span-2103">
              ⋆
             </span>
            </span>
            <span class="mo" id="MathJax-Span-2104">
             &gt;
            </span>
            <span class="mo" id="MathJax-Span-2105">
             −
            </span>
            <span class="mi" id="MathJax-Span-2106">
             ∞
            </span>
           </span>
          </span>
         </span>
        </span>
        时，存在一组对偶变量
        <span class="editor-math-element" data-id="JTI1MjQlMjUyOCUyNTVDbGFtYmRhJTI1NUUlMjU1Q3N0YXIlMjUyQyUyNTVDbnUlMjU1RSUyNTVDc3RhciUyNTI5JTI1MjQ=">
         <span class="MathJax" id="MathJax-Element-108-Frame">
          <span class="math" id="MathJax-Span-2107">
           <span class="mrow" id="MathJax-Span-2108">
            <span class="mo" id="MathJax-Span-2109">
             (
            </span>
            <span class="msubsup" id="MathJax-Span-2110">
             <span class="mi" id="MathJax-Span-2111">
              λ
             </span>
             <span class="mo" id="MathJax-Span-2112">
              ⋆
             </span>
            </span>
            <span class="mo" id="MathJax-Span-2113">
             ,
            </span>
            <span class="msubsup" id="MathJax-Span-2114">
             <span class="mi" id="MathJax-Span-2115">
              ν
             </span>
             <span class="mo" id="MathJax-Span-2116">
              ⋆
             </span>
            </span>
            <span class="mo" id="MathJax-Span-2117">
             )
            </span>
           </span>
          </span>
         </span>
        </span>
        满足
        <span class="editor-math-element" data-id="JTI1MjRnJTI1MjglMjU1Q2xhbWJkYSUyNTVFJTI1NUNzdGFyJTI1MkMlMjU1Q251JTI1NUUlMjU1Q3N0YXIlMjUyOSUyNTNEZCUyNTVFJTI1NUNzdGFyJTI1M0RwJTI1NUUlMjU1Q3N0YXIlMjUyNA==">
         <span class="MathJax" id="MathJax-Element-109-Frame">
          <span class="math" id="MathJax-Span-2118">
           <span class="mrow" id="MathJax-Span-2119">
            <span class="mi" id="MathJax-Span-2120">
             g
            </span>
            <span class="mo" id="MathJax-Span-2121">
             (
            </span>
            <span class="msubsup" id="MathJax-Span-2122">
             <span class="mi" id="MathJax-Span-2123">
              λ
             </span>
             <span class="mo" id="MathJax-Span-2124">
              ⋆
             </span>
            </span>
            <span class="mo" id="MathJax-Span-2125">
             ,
            </span>
            <span class="msubsup" id="MathJax-Span-2126">
             <span class="mi" id="MathJax-Span-2127">
              ν
             </span>
             <span class="mo" id="MathJax-Span-2128">
              ⋆
             </span>
            </span>
            <span class="mo" id="MathJax-Span-2129">
             )
            </span>
            <span class="mo" id="MathJax-Span-2130">
             =
            </span>
            <span class="msubsup" id="MathJax-Span-2131">
             <span class="mi" id="MathJax-Span-2132">
              d
             </span>
             <span class="mo" id="MathJax-Span-2133">
              ⋆
             </span>
            </span>
            <span class="mo" id="MathJax-Span-2134">
             =
            </span>
            <span class="msubsup" id="MathJax-Span-2135">
             <span class="mi" id="MathJax-Span-2136">
              p
             </span>
             <span class="mo" id="MathJax-Span-2137">
              ⋆
             </span>
            </span>
           </span>
          </span>
         </span>
        </span>
        ，即此时对偶最优值是可取到的。
       </p>
       <h2>
        原始问题与对偶问题的关系
        <span id="anchor_JTI1dTUzOUYlMjV1NTlDQiUyNXU5NUVFJTI1dTk4OTglMjV1NEUwRSUyNXU1QkY5JTI1dTUwNzYlMjV1OTVFRSUyNXU5ODk4JTI1dTc2ODQlMjV1NTE3MyUyNXU3Q0ZCMTM=">
        </span>
       </h2>
       <ol>
        <li>
         原始问题和对偶问题都是可行的，则弱对偶关系成立，强对偶关系不一定成立。
        </li>
        <li>
         原始问题和对偶问题都不可行，则弱对偶关系依然成立，但强对偶关系不成立。
        </li>
        <li>
         关于原始问题和对偶问题之间的解的关系，可以整理如下表格：
        </li>
       </ol>
       <table>
        <thead>
         <tr>
          <th>
           对偶问题\原始问题
          </th>
          <th>
           可行
          </th>
          <th>
           无下界
          </th>
          <th>
           不可行
          </th>
         </tr>
        </thead>
        <tbody>
         <tr>
          <td>
           可行
          </td>
          <td>
           √
          </td>
          <td>
           ×
          </td>
          <td>
           ×
          </td>
         </tr>
         <tr>
          <td>
           无上界
          </td>
          <td>
           ×
          </td>
          <td>
           ×
          </td>
          <td>
           √
          </td>
         </tr>
         <tr>
          <td>
           不可行
          </td>
          <td>
           ×
          </td>
          <td>
           √
          </td>
          <td>
           √
          </td>
         </tr>
        </tbody>
       </table>
       <h2>
        最优条件
        <span id="anchor_JTI1dTY3MDAlMjV1NEYxOCUyNXU2NzYxJTI1dTRFRjYxNA==">
        </span>
       </h2>
       <p>
        现在假设我们现在已经知道了原始问题和对偶问题的最优值相等（强对偶）。
        <span class="editor-math-element" data-id="JTI1MjR4JTI1NUUlMjU1Q2FzdCUyNTI0">
         <span class="MathJax" id="MathJax-Element-110-Frame">
          <span class="math" id="MathJax-Span-2138">
           <span class="mrow" id="MathJax-Span-2139">
            <span class="msubsup" id="MathJax-Span-2140">
             <span class="mi" id="MathJax-Span-2141">
              x
             </span>
             <span class="mo" id="MathJax-Span-2142">
              ∗
             </span>
            </span>
           </span>
          </span>
         </span>
        </span>
        是原始问题的最优解，
        <span class="editor-math-element" data-id="JTI1MjQlMjUyOCUyNTVDbGFtYmRhJTI1NUUlMjU1Q2FzdCUyNTJDJTI1MjAlMjU1Q251JTI1NUUlMjU1Q2FzdCUyNTI5JTI1MjQ=">
         <span class="MathJax" id="MathJax-Element-111-Frame">
          <span class="math" id="MathJax-Span-2143">
           <span class="mrow" id="MathJax-Span-2144">
            <span class="mo" id="MathJax-Span-2145">
             (
            </span>
            <span class="msubsup" id="MathJax-Span-2146">
             <span class="mi" id="MathJax-Span-2147">
              λ
             </span>
             <span class="mo" id="MathJax-Span-2148">
              ∗
             </span>
            </span>
            <span class="mo" id="MathJax-Span-2149">
             ,
            </span>
            <span class="msubsup" id="MathJax-Span-2150">
             <span class="mi" id="MathJax-Span-2151">
              ν
             </span>
             <span class="mo" id="MathJax-Span-2152">
              ∗
             </span>
            </span>
            <span class="mo" id="MathJax-Span-2153">
             )
            </span>
           </span>
          </span>
         </span>
        </span>
        是对偶问题的最优解。我们可以写出如下的式子：
       </p>
       <div class="MathJax_Display">
        <span class="MathJax" id="MathJax-Element-112-Frame">
         <span class="math" id="MathJax-Span-2154">
          <span class="mrow" id="MathJax-Span-2155">
           <span class="mtable" id="MathJax-Span-2156">
            <span class="mtd" id="MathJax-Span-2157">
             <span class="mrow" id="MathJax-Span-2158">
              <span class="msubsup" id="MathJax-Span-2159">
               <span class="mi" id="MathJax-Span-2160">
                f
               </span>
               <span class="mn" id="MathJax-Span-2161">
                0
               </span>
              </span>
              <span class="mo" id="MathJax-Span-2162">
               (
              </span>
              <span class="msubsup" id="MathJax-Span-2163">
               <span class="mi" id="MathJax-Span-2164">
                x
               </span>
               <span class="mo" id="MathJax-Span-2165">
                ∗
               </span>
              </span>
              <span class="mo" id="MathJax-Span-2166">
               )
              </span>
              <span class="mo" id="MathJax-Span-2167">
               =
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-2180">
             <span class="mrow" id="MathJax-Span-2181">
              <span class="mo" id="MathJax-Span-2182">
               =
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-2237">
             <span class="mrow" id="MathJax-Span-2238">
              <span class="mo" id="MathJax-Span-2239">
               ≤
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-2292">
             <span class="mrow" id="MathJax-Span-2293">
              <span class="mo" id="MathJax-Span-2294">
               ≤
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-2168">
             <span class="mrow" id="MathJax-Span-2169">
              <span class="mi" id="MathJax-Span-2170">
               g
              </span>
              <span class="mo" id="MathJax-Span-2171">
               (
              </span>
              <span class="msubsup" id="MathJax-Span-2172">
               <span class="mi" id="MathJax-Span-2173">
                λ
               </span>
               <span class="mo" id="MathJax-Span-2174">
                ∗
               </span>
              </span>
              <span class="mo" id="MathJax-Span-2175">
               ,
              </span>
              <span class="msubsup" id="MathJax-Span-2176">
               <span class="mi" id="MathJax-Span-2177">
                ν
               </span>
               <span class="mo" id="MathJax-Span-2178">
                ∗
               </span>
              </span>
              <span class="mo" id="MathJax-Span-2179">
               )
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-2183">
             <span class="mrow" id="MathJax-Span-2184">
              <span class="munderover" id="MathJax-Span-2185">
               <span class="mo" id="MathJax-Span-2186">
                inf
               </span>
               <span class="texatom" id="MathJax-Span-2187">
                <span class="mrow" id="MathJax-Span-2188">
                 <span class="mi" id="MathJax-Span-2189">
                  x
                 </span>
                </span>
               </span>
              </span>
              <span class="mrow" id="MathJax-Span-2190">
               <span class="mo" id="MathJax-Span-2191">
                (
               </span>
               <span class="msubsup" id="MathJax-Span-2192">
                <span class="mi" id="MathJax-Span-2193">
                 f
                </span>
                <span class="mn" id="MathJax-Span-2194">
                 0
                </span>
               </span>
               <span class="mo" id="MathJax-Span-2195">
                (
               </span>
               <span class="mi" id="MathJax-Span-2196">
                x
               </span>
               <span class="mo" id="MathJax-Span-2197">
                )
               </span>
               <span class="mo" id="MathJax-Span-2198">
                +
               </span>
               <span class="munderover" id="MathJax-Span-2199">
                <span class="mo" id="MathJax-Span-2200">
                 ∑
                </span>
                <span class="texatom" id="MathJax-Span-2201">
                 <span class="mrow" id="MathJax-Span-2202">
                  <span class="mi" id="MathJax-Span-2203">
                   i
                  </span>
                  <span class="mo" id="MathJax-Span-2204">
                   =
                  </span>
                  <span class="mn" id="MathJax-Span-2205">
                   1
                  </span>
                 </span>
                </span>
                <span class="mi" id="MathJax-Span-2206">
                 m
                </span>
               </span>
               <span class="msubsup" id="MathJax-Span-2207">
                <span class="mi" id="MathJax-Span-2208">
                 λ
                </span>
                <span class="mo" id="MathJax-Span-2209">
                 ∗
                </span>
                <span class="mi" id="MathJax-Span-2210">
                 i
                </span>
               </span>
               <span class="msubsup" id="MathJax-Span-2211">
                <span class="mi" id="MathJax-Span-2212">
                 f
                </span>
                <span class="mi" id="MathJax-Span-2213">
                 i
                </span>
               </span>
               <span class="mo" id="MathJax-Span-2214">
                (
               </span>
               <span class="mi" id="MathJax-Span-2215">
                x
               </span>
               <span class="mo" id="MathJax-Span-2216">
                )
               </span>
               <span class="mo" id="MathJax-Span-2217">
                +
               </span>
               <span class="munderover" id="MathJax-Span-2218">
                <span class="mo" id="MathJax-Span-2219">
                 ∑
                </span>
                <span class="texatom" id="MathJax-Span-2220">
                 <span class="mrow" id="MathJax-Span-2221">
                  <span class="mi" id="MathJax-Span-2222">
                   i
                  </span>
                  <span class="mo" id="MathJax-Span-2223">
                   =
                  </span>
                  <span class="mn" id="MathJax-Span-2224">
                   1
                  </span>
                 </span>
                </span>
                <span class="mi" id="MathJax-Span-2225">
                 p
                </span>
               </span>
               <span class="msubsup" id="MathJax-Span-2226">
                <span class="mi" id="MathJax-Span-2227">
                 ν
                </span>
                <span class="mo" id="MathJax-Span-2228">
                 ∗
                </span>
                <span class="mi" id="MathJax-Span-2229">
                 i
                </span>
               </span>
               <span class="msubsup" id="MathJax-Span-2230">
                <span class="mi" id="MathJax-Span-2231">
                 h
                </span>
                <span class="mi" id="MathJax-Span-2232">
                 i
                </span>
               </span>
               <span class="mo" id="MathJax-Span-2233">
                (
               </span>
               <span class="mi" id="MathJax-Span-2234">
                x
               </span>
               <span class="mo" id="MathJax-Span-2235">
                )
               </span>
               <span class="mo" id="MathJax-Span-2236">
                )
               </span>
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-2240">
             <span class="mrow" id="MathJax-Span-2241">
              <span class="msubsup" id="MathJax-Span-2242">
               <span class="mi" id="MathJax-Span-2243">
                f
               </span>
               <span class="mn" id="MathJax-Span-2244">
                0
               </span>
              </span>
              <span class="mo" id="MathJax-Span-2245">
               (
              </span>
              <span class="msubsup" id="MathJax-Span-2246">
               <span class="mi" id="MathJax-Span-2247">
                x
               </span>
               <span class="mo" id="MathJax-Span-2248">
                ∗
               </span>
              </span>
              <span class="mo" id="MathJax-Span-2249">
               )
              </span>
              <span class="mo" id="MathJax-Span-2250">
               +
              </span>
              <span class="munderover" id="MathJax-Span-2251">
               <span class="mo" id="MathJax-Span-2252">
                ∑
               </span>
               <span class="texatom" id="MathJax-Span-2253">
                <span class="mrow" id="MathJax-Span-2254">
                 <span class="mi" id="MathJax-Span-2255">
                  i
                 </span>
                 <span class="mo" id="MathJax-Span-2256">
                  =
                 </span>
                 <span class="mn" id="MathJax-Span-2257">
                  1
                 </span>
                </span>
               </span>
               <span class="mi" id="MathJax-Span-2258">
                m
               </span>
              </span>
              <span class="msubsup" id="MathJax-Span-2259">
               <span class="mi" id="MathJax-Span-2260">
                λ
               </span>
               <span class="mo" id="MathJax-Span-2261">
                ∗
               </span>
               <span class="mi" id="MathJax-Span-2262">
                i
               </span>
              </span>
              <span class="msubsup" id="MathJax-Span-2263">
               <span class="mi" id="MathJax-Span-2264">
                f
               </span>
               <span class="mi" id="MathJax-Span-2265">
                i
               </span>
              </span>
              <span class="mo" id="MathJax-Span-2266">
               (
              </span>
              <span class="msubsup" id="MathJax-Span-2267">
               <span class="mi" id="MathJax-Span-2268">
                x
               </span>
               <span class="mo" id="MathJax-Span-2269">
                ∗
               </span>
              </span>
              <span class="mo" id="MathJax-Span-2270">
               )
              </span>
              <span class="mo" id="MathJax-Span-2271">
               +
              </span>
              <span class="munderover" id="MathJax-Span-2272">
               <span class="mo" id="MathJax-Span-2273">
                ∑
               </span>
               <span class="texatom" id="MathJax-Span-2274">
                <span class="mrow" id="MathJax-Span-2275">
                 <span class="mi" id="MathJax-Span-2276">
                  i
                 </span>
                 <span class="mo" id="MathJax-Span-2277">
                  =
                 </span>
                 <span class="mn" id="MathJax-Span-2278">
                  1
                 </span>
                </span>
               </span>
               <span class="mi" id="MathJax-Span-2279">
                p
               </span>
              </span>
              <span class="msubsup" id="MathJax-Span-2280">
               <span class="mi" id="MathJax-Span-2281">
                ν
               </span>
               <span class="mo" id="MathJax-Span-2282">
                ∗
               </span>
               <span class="mi" id="MathJax-Span-2283">
                i
               </span>
              </span>
              <span class="msubsup" id="MathJax-Span-2284">
               <span class="mi" id="MathJax-Span-2285">
                h
               </span>
               <span class="mi" id="MathJax-Span-2286">
                i
               </span>
              </span>
              <span class="mo" id="MathJax-Span-2287">
               (
              </span>
              <span class="msubsup" id="MathJax-Span-2288">
               <span class="mi" id="MathJax-Span-2289">
                x
               </span>
               <span class="mo" id="MathJax-Span-2290">
                ∗
               </span>
              </span>
              <span class="mo" id="MathJax-Span-2291">
               )
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-2295">
             <span class="mrow" id="MathJax-Span-2296">
              <span class="msubsup" id="MathJax-Span-2297">
               <span class="mi" id="MathJax-Span-2298">
                f
               </span>
               <span class="mn" id="MathJax-Span-2299">
                0
               </span>
              </span>
              <span class="mo" id="MathJax-Span-2300">
               (
              </span>
              <span class="msubsup" id="MathJax-Span-2301">
               <span class="mi" id="MathJax-Span-2302">
                x
               </span>
               <span class="mo" id="MathJax-Span-2303">
                ∗
               </span>
              </span>
              <span class="mo" id="MathJax-Span-2304">
               )
              </span>
             </span>
            </span>
           </span>
           <span class="mtext" id="MathJax-Span-2305">
           </span>
          </span>
         </span>
        </span>
       </div>
       <p>
        强对偶存在的情况下，上面的不等号都要取到等号。
       </p>
       <p>
        这里有两个不等号，第三行取到等号的必要条件是：
       </p>
       <div class="MathJax_Display">
        <span class="MathJax" id="MathJax-Element-113-Frame">
         <span class="math" id="MathJax-Span-2306">
          <span class="mrow" id="MathJax-Span-2307">
           <span class="mi" id="MathJax-Span-2308">
            ∇
           </span>
           <span class="msubsup" id="MathJax-Span-2309">
            <span class="mi" id="MathJax-Span-2310">
             f
            </span>
            <span class="mn" id="MathJax-Span-2311">
             0
            </span>
           </span>
           <span class="mo" id="MathJax-Span-2312">
            (
           </span>
           <span class="msubsup" id="MathJax-Span-2313">
            <span class="mi" id="MathJax-Span-2314">
             x
            </span>
            <span class="mo" id="MathJax-Span-2315">
             ∗
            </span>
           </span>
           <span class="mo" id="MathJax-Span-2316">
            )
           </span>
           <span class="mo" id="MathJax-Span-2317">
            +
           </span>
           <span class="munderover" id="MathJax-Span-2318">
            <span class="mo" id="MathJax-Span-2319">
             ∑
            </span>
            <span class="texatom" id="MathJax-Span-2320">
             <span class="mrow" id="MathJax-Span-2321">
              <span class="mi" id="MathJax-Span-2322">
               i
              </span>
              <span class="mo" id="MathJax-Span-2323">
               =
              </span>
              <span class="mn" id="MathJax-Span-2324">
               1
              </span>
             </span>
            </span>
            <span class="mi" id="MathJax-Span-2325">
             m
            </span>
           </span>
           <span class="msubsup" id="MathJax-Span-2326">
            <span class="mi" id="MathJax-Span-2327">
             λ
            </span>
            <span class="mo" id="MathJax-Span-2328">
             ∗
            </span>
            <span class="mi" id="MathJax-Span-2329">
             i
            </span>
           </span>
           <span class="mi" id="MathJax-Span-2330">
            ∇
           </span>
           <span class="msubsup" id="MathJax-Span-2331">
            <span class="mi" id="MathJax-Span-2332">
             f
            </span>
            <span class="mi" id="MathJax-Span-2333">
             i
            </span>
           </span>
           <span class="mo" id="MathJax-Span-2334">
            (
           </span>
           <span class="msubsup" id="MathJax-Span-2335">
            <span class="mi" id="MathJax-Span-2336">
             x
            </span>
            <span class="mo" id="MathJax-Span-2337">
             ∗
            </span>
           </span>
           <span class="mo" id="MathJax-Span-2338">
            )
           </span>
           <span class="mo" id="MathJax-Span-2339">
            +
           </span>
           <span class="munderover" id="MathJax-Span-2340">
            <span class="mo" id="MathJax-Span-2341">
             ∑
            </span>
            <span class="texatom" id="MathJax-Span-2342">
             <span class="mrow" id="MathJax-Span-2343">
              <span class="mi" id="MathJax-Span-2344">
               i
              </span>
              <span class="mo" id="MathJax-Span-2345">
               =
              </span>
              <span class="mn" id="MathJax-Span-2346">
               1
              </span>
             </span>
            </span>
            <span class="mi" id="MathJax-Span-2347">
             p
            </span>
           </span>
           <span class="msubsup" id="MathJax-Span-2348">
            <span class="mi" id="MathJax-Span-2349">
             ν
            </span>
            <span class="mo" id="MathJax-Span-2350">
             ∗
            </span>
            <span class="mi" id="MathJax-Span-2351">
             i
            </span>
           </span>
           <span class="mi" id="MathJax-Span-2352">
            ∇
           </span>
           <span class="msubsup" id="MathJax-Span-2353">
            <span class="mi" id="MathJax-Span-2354">
             h
            </span>
            <span class="mi" id="MathJax-Span-2355">
             i
            </span>
           </span>
           <span class="mo" id="MathJax-Span-2356">
            (
           </span>
           <span class="msubsup" id="MathJax-Span-2357">
            <span class="mi" id="MathJax-Span-2358">
             x
            </span>
            <span class="mo" id="MathJax-Span-2359">
             ∗
            </span>
           </span>
           <span class="mo" id="MathJax-Span-2360">
            )
           </span>
           <span class="mo" id="MathJax-Span-2361">
            =
           </span>
           <span class="mn" id="MathJax-Span-2362">
            0
           </span>
           <span class="mtext" id="MathJax-Span-2363">
           </span>
          </span>
         </span>
        </span>
       </div>
       <p>
        这个条件会出现在下面的KKT条件中。
       </p>
       <p>
        而第四行取到等号的条件是：
       </p>
       <div class="MathJax_Display">
        <span class="MathJax" id="MathJax-Element-114-Frame">
         <span class="math" id="MathJax-Span-2364">
          <span class="mrow" id="MathJax-Span-2365">
           <span class="munderover" id="MathJax-Span-2366">
            <span class="mo" id="MathJax-Span-2367">
             ∑
            </span>
            <span class="texatom" id="MathJax-Span-2368">
             <span class="mrow" id="MathJax-Span-2369">
              <span class="mi" id="MathJax-Span-2370">
               i
              </span>
              <span class="mo" id="MathJax-Span-2371">
               =
              </span>
              <span class="mn" id="MathJax-Span-2372">
               1
              </span>
             </span>
            </span>
            <span class="mi" id="MathJax-Span-2373">
             m
            </span>
           </span>
           <span class="msubsup" id="MathJax-Span-2374">
            <span class="mi" id="MathJax-Span-2375">
             λ
            </span>
            <span class="mo" id="MathJax-Span-2376">
             ∗
            </span>
            <span class="mi" id="MathJax-Span-2377">
             i
            </span>
           </span>
           <span class="msubsup" id="MathJax-Span-2378">
            <span class="mi" id="MathJax-Span-2379">
             f
            </span>
            <span class="mi" id="MathJax-Span-2380">
             i
            </span>
           </span>
           <span class="mo" id="MathJax-Span-2381">
            (
           </span>
           <span class="msubsup" id="MathJax-Span-2382">
            <span class="mi" id="MathJax-Span-2383">
             x
            </span>
            <span class="mo" id="MathJax-Span-2384">
             ∗
            </span>
           </span>
           <span class="mo" id="MathJax-Span-2385">
            )
           </span>
           <span class="mo" id="MathJax-Span-2386">
            =
           </span>
           <span class="mn" id="MathJax-Span-2387">
            0
           </span>
           <span class="mtext" id="MathJax-Span-2388">
           </span>
          </span>
         </span>
        </span>
       </div>
       <p>
        这个条件可以得到下面的互补松弛（Complementary slackness）条件。
       </p>
       <h3>
        互补松弛条件
        <span id="anchor_JTI1dTRFOTIlMjV1ODg2NSUyNXU2NzdFJTI1dTVGMUIlMjV1Njc2MSUyNXU0RUY2MTU=">
        </span>
       </h3>
       <p>
        上面得到：
       </p>
       <div class="MathJax_Display">
        <span class="MathJax" id="MathJax-Element-115-Frame">
         <span class="math" id="MathJax-Span-2389">
          <span class="mrow" id="MathJax-Span-2390">
           <span class="munderover" id="MathJax-Span-2391">
            <span class="mo" id="MathJax-Span-2392">
             ∑
            </span>
            <span class="texatom" id="MathJax-Span-2393">
             <span class="mrow" id="MathJax-Span-2394">
              <span class="mi" id="MathJax-Span-2395">
               i
              </span>
              <span class="mo" id="MathJax-Span-2396">
               =
              </span>
              <span class="mn" id="MathJax-Span-2397">
               1
              </span>
             </span>
            </span>
            <span class="mi" id="MathJax-Span-2398">
             m
            </span>
           </span>
           <span class="msubsup" id="MathJax-Span-2399">
            <span class="mi" id="MathJax-Span-2400">
             λ
            </span>
            <span class="mo" id="MathJax-Span-2401">
             ∗
            </span>
            <span class="mi" id="MathJax-Span-2402">
             i
            </span>
           </span>
           <span class="msubsup" id="MathJax-Span-2403">
            <span class="mi" id="MathJax-Span-2404">
             f
            </span>
            <span class="mi" id="MathJax-Span-2405">
             i
            </span>
           </span>
           <span class="mo" id="MathJax-Span-2406">
            (
           </span>
           <span class="msubsup" id="MathJax-Span-2407">
            <span class="mi" id="MathJax-Span-2408">
             x
            </span>
            <span class="mo" id="MathJax-Span-2409">
             ∗
            </span>
           </span>
           <span class="mo" id="MathJax-Span-2410">
            )
           </span>
           <span class="mo" id="MathJax-Span-2411">
            =
           </span>
           <span class="mn" id="MathJax-Span-2412">
            0
           </span>
           <span class="mtext" id="MathJax-Span-2413">
           </span>
          </span>
         </span>
        </span>
       </div>
       <p>
        由于这个求和的每一项都小于等于0，所以我们可以得到：
       </p>
       <div class="MathJax_Display">
        <span class="MathJax" id="MathJax-Element-116-Frame">
         <span class="math" id="MathJax-Span-2414">
          <span class="mrow" id="MathJax-Span-2415">
           <span class="msubsup" id="MathJax-Span-2416">
            <span class="mi" id="MathJax-Span-2417">
             λ
            </span>
            <span class="mo" id="MathJax-Span-2418">
             ∗
            </span>
            <span class="mi" id="MathJax-Span-2419">
             i
            </span>
           </span>
           <span class="msubsup" id="MathJax-Span-2420">
            <span class="mi" id="MathJax-Span-2421">
             f
            </span>
            <span class="mi" id="MathJax-Span-2422">
             i
            </span>
           </span>
           <span class="mo" id="MathJax-Span-2423">
            (
           </span>
           <span class="msubsup" id="MathJax-Span-2424">
            <span class="mi" id="MathJax-Span-2425">
             x
            </span>
            <span class="mo" id="MathJax-Span-2426">
             ∗
            </span>
           </span>
           <span class="mo" id="MathJax-Span-2427">
            )
           </span>
           <span class="mo" id="MathJax-Span-2428">
            =
           </span>
           <span class="mn" id="MathJax-Span-2429">
            0
           </span>
           <span class="mo" id="MathJax-Span-2430">
            ,
           </span>
           <span class="mspace" id="MathJax-Span-2431">
           </span>
           <span class="mi" id="MathJax-Span-2432">
            i
           </span>
           <span class="mo" id="MathJax-Span-2433">
            =
           </span>
           <span class="mn" id="MathJax-Span-2434">
            1
           </span>
           <span class="mo" id="MathJax-Span-2435">
            ,
           </span>
           <span class="mo" id="MathJax-Span-2436">
            …
           </span>
           <span class="mo" id="MathJax-Span-2437">
            ,
           </span>
           <span class="mi" id="MathJax-Span-2438">
            m
           </span>
           <span class="mtext" id="MathJax-Span-2439">
           </span>
          </span>
         </span>
        </span>
       </div>
       <p>
        这就是著名的互补松弛条件。由这个条件我们可以得到如下结论：
       </p>
       <ol>
        <li>
         如果拉格朗日不等约束乘子
         <span class="editor-math-element" data-id="JTI1MjQlMjU1Q2xhbWJkYV9pJTI1NUUlMjU1Q2FzdCUyNTIwJTI1NUNndCUyNTIwMCUyNTI0">
          <span class="MathJax" id="MathJax-Element-117-Frame">
           <span class="math" id="MathJax-Span-2440">
            <span class="mrow" id="MathJax-Span-2441">
             <span class="msubsup" id="MathJax-Span-2442">
              <span class="mi" id="MathJax-Span-2443">
               λ
              </span>
              <span class="mo" id="MathJax-Span-2444">
               ∗
              </span>
              <span class="mi" id="MathJax-Span-2445">
               i
              </span>
             </span>
             <span class="mo" id="MathJax-Span-2446">
              &gt;
             </span>
             <span class="mn" id="MathJax-Span-2447">
              0
             </span>
            </span>
           </span>
          </span>
         </span>
         ，则该乘子对应的约束条件严格取等号，即
         <span class="editor-math-element" data-id="JTI1MjRmX2klMjUyOHglMjU1RSUyNTVDYXN0JTI1MjklMjUyMCUyNTNEJTI1MjAwJTI1MjQ=">
          <span class="MathJax" id="MathJax-Element-118-Frame">
           <span class="math" id="MathJax-Span-2448">
            <span class="mrow" id="MathJax-Span-2449">
             <span class="msubsup" id="MathJax-Span-2450">
              <span class="mi" id="MathJax-Span-2451">
               f
              </span>
              <span class="mi" id="MathJax-Span-2452">
               i
              </span>
             </span>
             <span class="mo" id="MathJax-Span-2453">
              (
             </span>
             <span class="msubsup" id="MathJax-Span-2454">
              <span class="mi" id="MathJax-Span-2455">
               x
              </span>
              <span class="mo" id="MathJax-Span-2456">
               ∗
              </span>
             </span>
             <span class="mo" id="MathJax-Span-2457">
              )
             </span>
             <span class="mo" id="MathJax-Span-2458">
              =
             </span>
             <span class="mn" id="MathJax-Span-2459">
              0
             </span>
            </span>
           </span>
          </span>
         </span>
         。
        </li>
        <li>
         如果某不等约束严格取不等号，即
         <span class="editor-math-element" data-id="JTI1MjRmX2klMjUyOHglMjU1RSUyNTVDYXN0JTI1MjklMjUyMCUyNTVDbHQlMjUyMDAlMjUyNA==">
          <span class="MathJax" id="MathJax-Element-119-Frame">
           <span class="math" id="MathJax-Span-2460">
            <span class="mrow" id="MathJax-Span-2461">
             <span class="msubsup" id="MathJax-Span-2462">
              <span class="mi" id="MathJax-Span-2463">
               f
              </span>
              <span class="mi" id="MathJax-Span-2464">
               i
              </span>
             </span>
             <span class="mo" id="MathJax-Span-2465">
              (
             </span>
             <span class="msubsup" id="MathJax-Span-2466">
              <span class="mi" id="MathJax-Span-2467">
               x
              </span>
              <span class="mo" id="MathJax-Span-2468">
               ∗
              </span>
             </span>
             <span class="mo" id="MathJax-Span-2469">
              )
             </span>
             <span class="mo" id="MathJax-Span-2470">
              &lt;
             </span>
             <span class="mn" id="MathJax-Span-2471">
              0
             </span>
            </span>
           </span>
          </span>
         </span>
         ，则该约束对应的乘子一定为零，即
         <span class="editor-math-element" data-id="JTI1MjQlMjU1Q2xhbWJkYV9pJTI1NUUlMjU1Q2FzdCUyNTIwJTI1M0QlMjUyMDAlMjUyNA==">
          <span class="MathJax" id="MathJax-Element-120-Frame">
           <span class="math" id="MathJax-Span-2472">
            <span class="mrow" id="MathJax-Span-2473">
             <span class="msubsup" id="MathJax-Span-2474">
              <span class="mi" id="MathJax-Span-2475">
               λ
              </span>
              <span class="mo" id="MathJax-Span-2476">
               ∗
              </span>
              <span class="mi" id="MathJax-Span-2477">
               i
              </span>
             </span>
             <span class="mo" id="MathJax-Span-2478">
              =
             </span>
             <span class="mn" id="MathJax-Span-2479">
              0
             </span>
            </span>
           </span>
          </span>
         </span>
         。
        </li>
       </ol>
       <p>
        互补松弛条件通常代表着一定物理意义。其中的乘子常常是一个明确的状态指示器。代表着约束的有效与否。
       </p>
       <h3>
        KKT条件
        <span id="anchor_S0tUJTI1dTY3NjElMjV1NEVGNjE2">
        </span>
       </h3>
       <h4>
        一般问题的KKT条件
        <span id="anchor_JTI1dTRFMDAlMjV1ODIyQyUyNXU5NUVFJTI1dTk4OTglMjV1NzY4NEtLVCUyNXU2NzYxJTI1dTRFRjYxNw==">
        </span>
       </h4>
       <p>
        将上面讨论的条件结合起来，我们就得到了著名的KKT条件：
       </p>
       <div class="MathJax_Display">
        <span class="MathJax" id="MathJax-Element-121-Frame">
         <span class="math" id="MathJax-Span-2480">
          <span class="mrow" id="MathJax-Span-2481">
           <span class="mtable" id="MathJax-Span-2482">
            <span class="mtd" id="MathJax-Span-2483">
             <span class="mrow" id="MathJax-Span-2484">
              <span class="msubsup" id="MathJax-Span-2485">
               <span class="mi" id="MathJax-Span-2486">
                f
               </span>
               <span class="mi" id="MathJax-Span-2487">
                i
               </span>
              </span>
              <span class="mo" id="MathJax-Span-2488">
               (
              </span>
              <span class="msubsup" id="MathJax-Span-2489">
               <span class="mi" id="MathJax-Span-2490">
                x
               </span>
               <span class="mo" id="MathJax-Span-2491">
                ∗
               </span>
              </span>
              <span class="mo" id="MathJax-Span-2492">
               )
              </span>
              <span class="mo" id="MathJax-Span-2493">
               ≤
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-2506">
             <span class="mrow" id="MathJax-Span-2507">
              <span class="msubsup" id="MathJax-Span-2508">
               <span class="mi" id="MathJax-Span-2509">
                h
               </span>
               <span class="mi" id="MathJax-Span-2510">
                i
               </span>
              </span>
              <span class="mo" id="MathJax-Span-2511">
               (
              </span>
              <span class="msubsup" id="MathJax-Span-2512">
               <span class="mi" id="MathJax-Span-2513">
                x
               </span>
               <span class="mo" id="MathJax-Span-2514">
                ∗
               </span>
              </span>
              <span class="mo" id="MathJax-Span-2515">
               )
              </span>
              <span class="mo" id="MathJax-Span-2516">
               =
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-2529">
             <span class="mrow" id="MathJax-Span-2530">
              <span class="msubsup" id="MathJax-Span-2531">
               <span class="mi" id="MathJax-Span-2532">
                λ
               </span>
               <span class="mo" id="MathJax-Span-2533">
                ∗
               </span>
               <span class="mi" id="MathJax-Span-2534">
                i
               </span>
              </span>
              <span class="mo" id="MathJax-Span-2535">
               ≥
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-2548">
             <span class="mrow" id="MathJax-Span-2549">
              <span class="msubsup" id="MathJax-Span-2550">
               <span class="mi" id="MathJax-Span-2551">
                λ
               </span>
               <span class="mo" id="MathJax-Span-2552">
                ∗
               </span>
               <span class="mi" id="MathJax-Span-2553">
                i
               </span>
              </span>
              <span class="msubsup" id="MathJax-Span-2554">
               <span class="mi" id="MathJax-Span-2555">
                f
               </span>
               <span class="mi" id="MathJax-Span-2556">
                i
               </span>
              </span>
              <span class="mo" id="MathJax-Span-2557">
               (
              </span>
              <span class="msubsup" id="MathJax-Span-2558">
               <span class="mi" id="MathJax-Span-2559">
                x
               </span>
               <span class="mo" id="MathJax-Span-2560">
                ∗
               </span>
              </span>
              <span class="mo" id="MathJax-Span-2561">
               )
              </span>
              <span class="mo" id="MathJax-Span-2562">
               =
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-2575">
             <span class="mrow" id="MathJax-Span-2576">
              <span class="mi" id="MathJax-Span-2577">
               ∇
              </span>
              <span class="msubsup" id="MathJax-Span-2578">
               <span class="mi" id="MathJax-Span-2579">
                f
               </span>
               <span class="mn" id="MathJax-Span-2580">
                0
               </span>
              </span>
              <span class="mo" id="MathJax-Span-2581">
               (
              </span>
              <span class="msubsup" id="MathJax-Span-2582">
               <span class="mi" id="MathJax-Span-2583">
                x
               </span>
               <span class="mo" id="MathJax-Span-2584">
                ∗
               </span>
              </span>
              <span class="mo" id="MathJax-Span-2585">
               )
              </span>
              <span class="mo" id="MathJax-Span-2586">
               +
              </span>
              <span class="munderover" id="MathJax-Span-2587">
               <span class="mo" id="MathJax-Span-2588">
                ∑
               </span>
               <span class="texatom" id="MathJax-Span-2589">
                <span class="mrow" id="MathJax-Span-2590">
                 <span class="mi" id="MathJax-Span-2591">
                  i
                 </span>
                 <span class="mo" id="MathJax-Span-2592">
                  =
                 </span>
                 <span class="mn" id="MathJax-Span-2593">
                  1
                 </span>
                </span>
               </span>
               <span class="mi" id="MathJax-Span-2594">
                m
               </span>
              </span>
              <span class="msubsup" id="MathJax-Span-2595">
               <span class="mi" id="MathJax-Span-2596">
                λ
               </span>
               <span class="mo" id="MathJax-Span-2597">
                ∗
               </span>
               <span class="mi" id="MathJax-Span-2598">
                i
               </span>
              </span>
              <span class="mi" id="MathJax-Span-2599">
               ∇
              </span>
              <span class="msubsup" id="MathJax-Span-2600">
               <span class="mi" id="MathJax-Span-2601">
                f
               </span>
               <span class="mi" id="MathJax-Span-2602">
                i
               </span>
              </span>
              <span class="mo" id="MathJax-Span-2603">
               (
              </span>
              <span class="msubsup" id="MathJax-Span-2604">
               <span class="mi" id="MathJax-Span-2605">
                x
               </span>
               <span class="mo" id="MathJax-Span-2606">
                ∗
               </span>
              </span>
              <span class="mo" id="MathJax-Span-2607">
               )
              </span>
              <span class="mo" id="MathJax-Span-2608">
               +
              </span>
              <span class="munderover" id="MathJax-Span-2609">
               <span class="mo" id="MathJax-Span-2610">
                ∑
               </span>
               <span class="texatom" id="MathJax-Span-2611">
                <span class="mrow" id="MathJax-Span-2612">
                 <span class="mi" id="MathJax-Span-2613">
                  i
                 </span>
                 <span class="mo" id="MathJax-Span-2614">
                  =
                 </span>
                 <span class="mn" id="MathJax-Span-2615">
                  1
                 </span>
                </span>
               </span>
               <span class="mi" id="MathJax-Span-2616">
                p
               </span>
              </span>
              <span class="msubsup" id="MathJax-Span-2617">
               <span class="mi" id="MathJax-Span-2618">
                ν
               </span>
               <span class="mo" id="MathJax-Span-2619">
                ∗
               </span>
               <span class="mi" id="MathJax-Span-2620">
                i
               </span>
              </span>
              <span class="mi" id="MathJax-Span-2621">
               ∇
              </span>
              <span class="msubsup" id="MathJax-Span-2622">
               <span class="mi" id="MathJax-Span-2623">
                h
               </span>
               <span class="mi" id="MathJax-Span-2624">
                i
               </span>
              </span>
              <span class="mo" id="MathJax-Span-2625">
               (
              </span>
              <span class="msubsup" id="MathJax-Span-2626">
               <span class="mi" id="MathJax-Span-2627">
                x
               </span>
               <span class="mo" id="MathJax-Span-2628">
                ∗
               </span>
              </span>
              <span class="mo" id="MathJax-Span-2629">
               )
              </span>
              <span class="mo" id="MathJax-Span-2630">
               =
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-2494">
             <span class="mrow" id="MathJax-Span-2495">
              <span class="mn" id="MathJax-Span-2496">
               0
              </span>
              <span class="mo" id="MathJax-Span-2497">
               ,
              </span>
              <span class="mspace" id="MathJax-Span-2498">
              </span>
              <span class="mi" id="MathJax-Span-2499">
               i
              </span>
              <span class="mo" id="MathJax-Span-2500">
               =
              </span>
              <span class="mn" id="MathJax-Span-2501">
               1
              </span>
              <span class="mo" id="MathJax-Span-2502">
               ,
              </span>
              <span class="mo" id="MathJax-Span-2503">
               …
              </span>
              <span class="mo" id="MathJax-Span-2504">
               ,
              </span>
              <span class="mi" id="MathJax-Span-2505">
               m
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-2517">
             <span class="mrow" id="MathJax-Span-2518">
              <span class="mn" id="MathJax-Span-2519">
               0
              </span>
              <span class="mo" id="MathJax-Span-2520">
               ,
              </span>
              <span class="mspace" id="MathJax-Span-2521">
              </span>
              <span class="mi" id="MathJax-Span-2522">
               i
              </span>
              <span class="mo" id="MathJax-Span-2523">
               =
              </span>
              <span class="mn" id="MathJax-Span-2524">
               1
              </span>
              <span class="mo" id="MathJax-Span-2525">
               ,
              </span>
              <span class="mo" id="MathJax-Span-2526">
               …
              </span>
              <span class="mo" id="MathJax-Span-2527">
               ,
              </span>
              <span class="mi" id="MathJax-Span-2528">
               p
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-2536">
             <span class="mrow" id="MathJax-Span-2537">
              <span class="mn" id="MathJax-Span-2538">
               0
              </span>
              <span class="mo" id="MathJax-Span-2539">
               ,
              </span>
              <span class="mspace" id="MathJax-Span-2540">
              </span>
              <span class="mi" id="MathJax-Span-2541">
               i
              </span>
              <span class="mo" id="MathJax-Span-2542">
               =
              </span>
              <span class="mn" id="MathJax-Span-2543">
               1
              </span>
              <span class="mo" id="MathJax-Span-2544">
               ,
              </span>
              <span class="mo" id="MathJax-Span-2545">
               …
              </span>
              <span class="mo" id="MathJax-Span-2546">
               ,
              </span>
              <span class="mi" id="MathJax-Span-2547">
               m
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-2563">
             <span class="mrow" id="MathJax-Span-2564">
              <span class="mn" id="MathJax-Span-2565">
               0
              </span>
              <span class="mo" id="MathJax-Span-2566">
               ,
              </span>
              <span class="mspace" id="MathJax-Span-2567">
              </span>
              <span class="mi" id="MathJax-Span-2568">
               i
              </span>
              <span class="mo" id="MathJax-Span-2569">
               =
              </span>
              <span class="mn" id="MathJax-Span-2570">
               1
              </span>
              <span class="mo" id="MathJax-Span-2571">
               ,
              </span>
              <span class="mo" id="MathJax-Span-2572">
               …
              </span>
              <span class="mo" id="MathJax-Span-2573">
               ,
              </span>
              <span class="mi" id="MathJax-Span-2574">
               m
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-2631">
             <span class="mrow" id="MathJax-Span-2632">
              <span class="mn" id="MathJax-Span-2633">
               0
              </span>
             </span>
            </span>
           </span>
           <span class="mtext" id="MathJax-Span-2634">
           </span>
          </span>
         </span>
        </span>
       </div>
       <p>
        这里前两行条件是原始问题的约束，第三行是拉格朗日函数的要求，第四行是互补松弛条件，最后一行是对拉格朗日函数取极值时候带来的一个必要条件。
       </p>
       <p>
        容易看出，由于最后一个条件的限制，对于任意优化问题，只要
        <span class="editor-math-element" data-id="JTI1MjR4JTI1NUUlMjU1Q2FzdCUyNTI0">
         <span class="MathJax" id="MathJax-Element-122-Frame">
          <span class="math" id="MathJax-Span-2635">
           <span class="mrow" id="MathJax-Span-2636">
            <span class="msubsup" id="MathJax-Span-2637">
             <span class="mi" id="MathJax-Span-2638">
              x
             </span>
             <span class="mo" id="MathJax-Span-2639">
              ∗
             </span>
            </span>
           </span>
          </span>
         </span>
        </span>
        和
        <span class="editor-math-element" data-id="JTI1MjQlMjUyOCUyNTVDbGFtYmRhJTI1NUUlMjU1Q2FzdCUyNTJDJTI1MjAlMjU1Q251JTI1NUUlMjU1Q2FzdCUyNTI5JTI1MjQ=">
         <span class="MathJax" id="MathJax-Element-123-Frame">
          <span class="math" id="MathJax-Span-2640">
           <span class="mrow" id="MathJax-Span-2641">
            <span class="mo" id="MathJax-Span-2642">
             (
            </span>
            <span class="msubsup" id="MathJax-Span-2643">
             <span class="mi" id="MathJax-Span-2644">
              λ
             </span>
             <span class="mo" id="MathJax-Span-2645">
              ∗
             </span>
            </span>
            <span class="mo" id="MathJax-Span-2646">
             ,
            </span>
            <span class="msubsup" id="MathJax-Span-2647">
             <span class="mi" id="MathJax-Span-2648">
              ν
             </span>
             <span class="mo" id="MathJax-Span-2649">
              ∗
             </span>
            </span>
            <span class="mo" id="MathJax-Span-2650">
             )
            </span>
           </span>
          </span>
         </span>
        </span>
        是原始问题和对偶问题的最优解，则其一定满足KKT条件。即KKT条件是一组解成为最优解的
        <strong>
         必要条件
        </strong>
        。
       </p>
       <h4>
        凸问题的KKT条件
        <span id="anchor_JTI1dTUxRjglMjV1OTVFRSUyNXU5ODk4JTI1dTc2ODRLS1QlMjV1Njc2MSUyNXU0RUY2MTg=">
        </span>
       </h4>
       <p>
        如果原始问题是凸的，则KKT条件也是充分的。这是因为KKT的最后一个条件在对拉格朗日函数取下确界的时候成为了充要条件。这时候我们有如下结论：
       </p>
       <ul>
        <li>
         如果一个凸优化问题有可微的目标函数和约束，并且满足Slater条件，则KKT条件是取得最优的充要条件：Slater条件保证了最优对偶间隙为零并且最优点可以取到；在此基础上
         <span class="editor-math-element" data-id="JTI1MjR4JTI1MjQ=">
          <span class="MathJax" id="MathJax-Element-124-Frame">
           <span class="math" id="MathJax-Span-2651">
            <span class="mrow" id="MathJax-Span-2652">
             <span class="mi" id="MathJax-Span-2653">
              x
             </span>
            </span>
           </span>
          </span>
         </span>
         是最优当且仅当
         <span class="editor-math-element" data-id="JTI1MjQlMjUyOHglMjUyQyUyNTVDbGFtYmRhJTI1MkMlMjU1Q251JTI1MjklMjUyNA==">
          <span class="MathJax" id="MathJax-Element-125-Frame">
           <span class="math" id="MathJax-Span-2654">
            <span class="mrow" id="MathJax-Span-2655">
             <span class="mo" id="MathJax-Span-2656">
              (
             </span>
             <span class="mi" id="MathJax-Span-2657">
              x
             </span>
             <span class="mo" id="MathJax-Span-2658">
              ,
             </span>
             <span class="mi" id="MathJax-Span-2659">
              λ
             </span>
             <span class="mo" id="MathJax-Span-2660">
              ,
             </span>
             <span class="mi" id="MathJax-Span-2661">
              ν
             </span>
             <span class="mo" id="MathJax-Span-2662">
              )
             </span>
            </span>
           </span>
          </span>
         </span>
         满足KKT条件。
        </li>
       </ul>
       <h4>
        KKT条件的用途
        <span id="anchor_S0tUJTI1dTY3NjElMjV1NEVGNiUyNXU3Njg0JTI1dTc1MjglMjV1OTAxNDE5">
        </span>
       </h4>
       <p>
        KKT条件在优化问题中有重要意义。它可以用于如下方面：
       </p>
       <ol>
        <li>
         有时候可以直接从KKT条件里得到最优的解析解。
        </li>
        <li>
         等式约束的优化问题，可以通过KKT条件转化为无约束方程求零点问题。
        </li>
        <li>
         有不等式约束的优化问题，可以使用KKT条件来简化，帮助求解。
        </li>
       </ol>
       <h1>
        拉格朗日乘数法的形象化解读
        <span id="anchor_JTI1dTYyQzklMjV1NjgzQyUyNXU2NzE3JTI1dTY1RTUlMjV1NEU1OCUyNXU2NTcwJTI1dTZDRDUlMjV1NzY4NCUyNXU1RjYyJTI1dThDNjElMjV1NTMxNiUyNXU4OUUzJTI1dThCRkIyMA==">
        </span>
       </h1>
       <p>
        上面的论述都是拉格朗日乘子法的数学基础。但是上面的公式无法解释一个问题：为何要如此构造拉格朗日函数？其背后的意义是什么？这一部分就试图来回答这个问题。
       </p>
       <h2>
        等式约束的拉格朗日乘子法
        <span id="anchor_JTI1dTdCNDklMjV1NUYwRiUyNXU3RUE2JTI1dTY3NUYlMjV1NzY4NCUyNXU2MkM5JTI1dTY4M0MlMjV1NjcxNyUyNXU2NUU1JTI1dTRFNTglMjV1NUI1MCUyNXU2Q0Q1MjE=">
        </span>
       </h2>
       <p>
        考虑这个决策变量是二维平面内点
        <span class="editor-math-element" data-id="JTI1MjQlMjUyOHglMjUyQ3klMjUyOSUyNTI0">
         <span class="MathJax" id="MathJax-Element-126-Frame">
          <span class="math" id="MathJax-Span-2663">
           <span class="mrow" id="MathJax-Span-2664">
            <span class="mo" id="MathJax-Span-2665">
             (
            </span>
            <span class="mi" id="MathJax-Span-2666">
             x
            </span>
            <span class="mo" id="MathJax-Span-2667">
             ,
            </span>
            <span class="mi" id="MathJax-Span-2668">
             y
            </span>
            <span class="mo" id="MathJax-Span-2669">
             )
            </span>
           </span>
          </span>
         </span>
        </span>
        的优化问题：
       </p>
       <div class="MathJax_Display">
        <span class="MathJax" id="MathJax-Element-127-Frame">
         <span class="math" id="MathJax-Span-2670">
          <span class="mrow" id="MathJax-Span-2671">
           <span class="mtable" id="MathJax-Span-2672">
            <span class="mtd" id="MathJax-Span-2673">
             <span class="mrow" id="MathJax-Span-2674">
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-2688">
             <span class="mrow" id="MathJax-Span-2689">
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-2675">
             <span class="mrow" id="MathJax-Span-2676">
              <span class="mtext" id="MathJax-Span-2677">
               minimize
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-2690">
             <span class="mrow" id="MathJax-Span-2691">
              <span class="mtext" id="MathJax-Span-2692">
               subject to
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-2678">
             <span class="mrow" id="MathJax-Span-2679">
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-2693">
             <span class="mrow" id="MathJax-Span-2694">
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-2680">
             <span class="mrow" id="MathJax-Span-2681">
              <span class="mi" id="MathJax-Span-2682">
               f
              </span>
              <span class="mo" id="MathJax-Span-2683">
               (
              </span>
              <span class="mi" id="MathJax-Span-2684">
               x
              </span>
              <span class="mo" id="MathJax-Span-2685">
               ,
              </span>
              <span class="mi" id="MathJax-Span-2686">
               y
              </span>
              <span class="mo" id="MathJax-Span-2687">
               )
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-2695">
             <span class="mrow" id="MathJax-Span-2696">
              <span class="mi" id="MathJax-Span-2697">
               g
              </span>
              <span class="mo" id="MathJax-Span-2698">
               (
              </span>
              <span class="mi" id="MathJax-Span-2699">
               x
              </span>
              <span class="mo" id="MathJax-Span-2700">
               ,
              </span>
              <span class="mi" id="MathJax-Span-2701">
               y
              </span>
              <span class="mo" id="MathJax-Span-2702">
               )
              </span>
              <span class="mo" id="MathJax-Span-2703">
               =
              </span>
              <span class="mi" id="MathJax-Span-2704">
               c
              </span>
             </span>
            </span>
           </span>
           <span class="mtext" id="MathJax-Span-2705">
           </span>
          </span>
         </span>
        </span>
       </div>
       <p>
        我们在二维平面内画出两个函数的图像。由于缺少第三维，我们使用等高线来表示目标函数
        <span class="editor-math-element" data-id="JTI1MjRmJTI1Mjh4JTI1MkN5JTI1MjklMjUyNA==">
         <span class="MathJax" id="MathJax-Element-128-Frame">
          <span class="math" id="MathJax-Span-2706">
           <span class="mrow" id="MathJax-Span-2707">
            <span class="mi" id="MathJax-Span-2708">
             f
            </span>
            <span class="mo" id="MathJax-Span-2709">
             (
            </span>
            <span class="mi" id="MathJax-Span-2710">
             x
            </span>
            <span class="mo" id="MathJax-Span-2711">
             ,
            </span>
            <span class="mi" id="MathJax-Span-2712">
             y
            </span>
            <span class="mo" id="MathJax-Span-2713">
             )
            </span>
           </span>
          </span>
         </span>
        </span>
        的函数值。如下图：
        <br/>
        <a href="http://dataunion.org/wp-content/uploads/2015/01/221.png">
         <img src="http://dataunion.org/wp-content/uploads/2015/01/221.png"/>
        </a>
        <br/>
        图中画出了两条
        <span class="editor-math-element" data-id="JTI1MjRmJTI1Mjh4JTI1MkN5JTI1MjklMjUyNA==">
         <span class="MathJax" id="MathJax-Element-129-Frame">
          <span class="math" id="MathJax-Span-2714">
           <span class="mrow" id="MathJax-Span-2715">
            <span class="mi" id="MathJax-Span-2716">
             f
            </span>
            <span class="mo" id="MathJax-Span-2717">
             (
            </span>
            <span class="mi" id="MathJax-Span-2718">
             x
            </span>
            <span class="mo" id="MathJax-Span-2719">
             ,
            </span>
            <span class="mi" id="MathJax-Span-2720">
             y
            </span>
            <span class="mo" id="MathJax-Span-2721">
             )
            </span>
           </span>
          </span>
         </span>
        </span>
        的等高线，黑色箭头则是等高线上的梯度
        <span class="editor-math-element" data-id="JTI1MjQlMjU1Q25hYmxhJTI1MjBmJTI1Mjh4JTI1MkN5JTI1MjklMjUyNA==">
         <span class="MathJax" id="MathJax-Element-130-Frame">
          <span class="math" id="MathJax-Span-2722">
           <span class="mrow" id="MathJax-Span-2723">
            <span class="mi" id="MathJax-Span-2724">
             ∇
            </span>
            <span class="mi" id="MathJax-Span-2725">
             f
            </span>
            <span class="mo" id="MathJax-Span-2726">
             (
            </span>
            <span class="mi" id="MathJax-Span-2727">
             x
            </span>
            <span class="mo" id="MathJax-Span-2728">
             ,
            </span>
            <span class="mi" id="MathJax-Span-2729">
             y
            </span>
            <span class="mo" id="MathJax-Span-2730">
             )
            </span>
           </span>
          </span>
         </span>
        </span>
        。从梯度的方向上来看，显然有
        <span class="editor-math-element" data-id="JTI1MjRkXzElMjUzRWRfMiUyNTI0">
         <span class="MathJax" id="MathJax-Element-131-Frame">
          <span class="math" id="MathJax-Span-2731">
           <span class="mrow" id="MathJax-Span-2732">
            <span class="msubsup" id="MathJax-Span-2733">
             <span class="mi" id="MathJax-Span-2734">
              d
             </span>
             <span class="mn" id="MathJax-Span-2735">
              1
             </span>
            </span>
            <span class="mo" id="MathJax-Span-2736">
             &gt;
            </span>
            <span class="msubsup" id="MathJax-Span-2737">
             <span class="mi" id="MathJax-Span-2738">
              d
             </span>
             <span class="mn" id="MathJax-Span-2739">
              2
             </span>
            </span>
           </span>
          </span>
         </span>
        </span>
        。绿色的线是约束，也就是说，只要正好落在这条绿线上的点才可能是满足要求的点。如果没有这条约束，
        <span class="editor-math-element" data-id="JTI1MjRmJTI1Mjh4JTI1MkN5JTI1MjklMjUyNA==">
         <span class="MathJax" id="MathJax-Element-132-Frame">
          <span class="math" id="MathJax-Span-2740">
           <span class="mrow" id="MathJax-Span-2741">
            <span class="mi" id="MathJax-Span-2742">
             f
            </span>
            <span class="mo" id="MathJax-Span-2743">
             (
            </span>
            <span class="mi" id="MathJax-Span-2744">
             x
            </span>
            <span class="mo" id="MathJax-Span-2745">
             ,
            </span>
            <span class="mi" id="MathJax-Span-2746">
             y
            </span>
            <span class="mo" id="MathJax-Span-2747">
             )
            </span>
           </span>
          </span>
         </span>
        </span>
        的最小值应该会落在最小那圈等高线内部的某一点上。而现在加上了约束，最小值点应该在哪里呢？显然应该是在
        <span class="editor-math-element" data-id="JTI1MjRmJTI1Mjh4JTI1MkN5JTI1MjklMjUyNA==">
         <span class="MathJax" id="MathJax-Element-133-Frame">
          <span class="math" id="MathJax-Span-2748">
           <span class="mrow" id="MathJax-Span-2749">
            <span class="mi" id="MathJax-Span-2750">
             f
            </span>
            <span class="mo" id="MathJax-Span-2751">
             (
            </span>
            <span class="mi" id="MathJax-Span-2752">
             x
            </span>
            <span class="mo" id="MathJax-Span-2753">
             ,
            </span>
            <span class="mi" id="MathJax-Span-2754">
             y
            </span>
            <span class="mo" id="MathJax-Span-2755">
             )
            </span>
           </span>
          </span>
         </span>
        </span>
        的等高线正好和约束线相切的位置。如果我们对约束也求梯度
        <span class="editor-math-element" data-id="JTI1MjQlMjU1Q25hYmxhJTI1MjBnJTI1Mjh4JTI1MkN5JTI1MjklMjUyNA==">
         <span class="MathJax" id="MathJax-Element-134-Frame">
          <span class="math" id="MathJax-Span-2756">
           <span class="mrow" id="MathJax-Span-2757">
            <span class="mi" id="MathJax-Span-2758">
             ∇
            </span>
            <span class="mi" id="MathJax-Span-2759">
             g
            </span>
            <span class="mo" id="MathJax-Span-2760">
             (
            </span>
            <span class="mi" id="MathJax-Span-2761">
             x
            </span>
            <span class="mo" id="MathJax-Span-2762">
             ,
            </span>
            <span class="mi" id="MathJax-Span-2763">
             y
            </span>
            <span class="mo" id="MathJax-Span-2764">
             )
            </span>
           </span>
          </span>
         </span>
        </span>
        ，则其梯度如图中绿色箭头所示。很容易看出来，要想让目标函数
        <span class="editor-math-element" data-id="JTI1MjRmJTI1Mjh4JTI1MkN5JTI1MjklMjUyNA==">
         <span class="MathJax" id="MathJax-Element-135-Frame">
          <span class="math" id="MathJax-Span-2765">
           <span class="mrow" id="MathJax-Span-2766">
            <span class="mi" id="MathJax-Span-2767">
             f
            </span>
            <span class="mo" id="MathJax-Span-2768">
             (
            </span>
            <span class="mi" id="MathJax-Span-2769">
             x
            </span>
            <span class="mo" id="MathJax-Span-2770">
             ,
            </span>
            <span class="mi" id="MathJax-Span-2771">
             y
            </span>
            <span class="mo" id="MathJax-Span-2772">
             )
            </span>
           </span>
          </span>
         </span>
        </span>
        的等高线和约束相切，则他们切点的梯度一定在一条直线上，即：
       </p>
       <div class="MathJax_Display">
        <span class="MathJax" id="MathJax-Element-136-Frame">
         <span class="math" id="MathJax-Span-2773">
          <span class="mrow" id="MathJax-Span-2774">
           <span class="mi" id="MathJax-Span-2775">
            ∇
           </span>
           <span class="mi" id="MathJax-Span-2776">
            f
           </span>
           <span class="mo" id="MathJax-Span-2777">
            (
           </span>
           <span class="mi" id="MathJax-Span-2778">
            x
           </span>
           <span class="mo" id="MathJax-Span-2779">
            ,
           </span>
           <span class="mi" id="MathJax-Span-2780">
            y
           </span>
           <span class="mo" id="MathJax-Span-2781">
            )
           </span>
           <span class="mo" id="MathJax-Span-2782">
            =
           </span>
           <span class="mi" id="MathJax-Span-2783">
            ν
           </span>
           <span class="mi" id="MathJax-Span-2784">
            ∇
           </span>
           <span class="mi" id="MathJax-Span-2785">
            g
           </span>
           <span class="mo" id="MathJax-Span-2786">
            (
           </span>
           <span class="mi" id="MathJax-Span-2787">
            x
           </span>
           <span class="mo" id="MathJax-Span-2788">
            ,
           </span>
           <span class="mi" id="MathJax-Span-2789">
            y
           </span>
           <span class="mo" id="MathJax-Span-2790">
            )
           </span>
           <span class="mtext" id="MathJax-Span-2791">
           </span>
          </span>
         </span>
        </span>
       </div>
       <p>
        其中
        <span class="editor-math-element" data-id="JTI1MjQlMjU1Q251JTI1MjQ=">
         <span class="MathJax" id="MathJax-Element-137-Frame">
          <span class="math" id="MathJax-Span-2792">
           <span class="mrow" id="MathJax-Span-2793">
            <span class="mi" id="MathJax-Span-2794">
             ν
            </span>
           </span>
          </span>
         </span>
        </span>
        可以是任何实数。
       </p>
       <p>
        因此，我们通过观察可以得到优化取到最小值的条件：
       </p>
       <div class="MathJax_Display">
        <span class="MathJax" id="MathJax-Element-138-Frame">
         <span class="math" id="MathJax-Span-2795">
          <span class="mrow" id="MathJax-Span-2796">
           <span class="mtable" id="MathJax-Span-2797">
            <span class="mtd" id="MathJax-Span-2798">
             <span class="mrow" id="MathJax-Span-2799">
              <span class="mi" id="MathJax-Span-2800">
               g
              </span>
              <span class="mo" id="MathJax-Span-2801">
               (
              </span>
              <span class="mi" id="MathJax-Span-2802">
               x
              </span>
              <span class="mo" id="MathJax-Span-2803">
               ,
              </span>
              <span class="mi" id="MathJax-Span-2804">
               y
              </span>
              <span class="mo" id="MathJax-Span-2805">
               )
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-2810">
             <span class="mrow" id="MathJax-Span-2811">
              <span class="mi" id="MathJax-Span-2812">
               ∇
              </span>
              <span class="mi" id="MathJax-Span-2813">
               f
              </span>
              <span class="mo" id="MathJax-Span-2814">
               (
              </span>
              <span class="mi" id="MathJax-Span-2815">
               x
              </span>
              <span class="mo" id="MathJax-Span-2816">
               ,
              </span>
              <span class="mi" id="MathJax-Span-2817">
               y
              </span>
              <span class="mo" id="MathJax-Span-2818">
               )
              </span>
              <span class="mo" id="MathJax-Span-2819">
               −
              </span>
              <span class="mi" id="MathJax-Span-2820">
               ν
              </span>
              <span class="mi" id="MathJax-Span-2821">
               ∇
              </span>
              <span class="mi" id="MathJax-Span-2822">
               g
              </span>
              <span class="mo" id="MathJax-Span-2823">
               (
              </span>
              <span class="mi" id="MathJax-Span-2824">
               x
              </span>
              <span class="mo" id="MathJax-Span-2825">
               ,
              </span>
              <span class="mi" id="MathJax-Span-2826">
               y
              </span>
              <span class="mo" id="MathJax-Span-2827">
               )
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-2806">
             <span class="mrow" id="MathJax-Span-2807">
              <span class="mo" id="MathJax-Span-2808">
               =
              </span>
              <span class="mi" id="MathJax-Span-2809">
               c
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-2828">
             <span class="mrow" id="MathJax-Span-2829">
              <span class="mo" id="MathJax-Span-2830">
               =
              </span>
              <span class="mn" id="MathJax-Span-2831">
               0
              </span>
             </span>
            </span>
           </span>
           <span class="mtext" id="MathJax-Span-2832">
           </span>
          </span>
         </span>
        </span>
       </div>
       <p>
        最为对比，我们使用前一部分推导的KKT条件来直接写出这个问题的KKT条件：
       </p>
       <div class="MathJax_Display">
        <span class="MathJax" id="MathJax-Element-139-Frame">
         <span class="math" id="MathJax-Span-2833">
          <span class="mrow" id="MathJax-Span-2834">
           <span class="mtable" id="MathJax-Span-2835">
            <span class="mtd" id="MathJax-Span-2836">
             <span class="mrow" id="MathJax-Span-2837">
              <span class="mi" id="MathJax-Span-2838">
               g
              </span>
              <span class="mo" id="MathJax-Span-2839">
               (
              </span>
              <span class="mi" id="MathJax-Span-2840">
               x
              </span>
              <span class="mo" id="MathJax-Span-2841">
               ,
              </span>
              <span class="mi" id="MathJax-Span-2842">
               y
              </span>
              <span class="mo" id="MathJax-Span-2843">
               )
              </span>
              <span class="mo" id="MathJax-Span-2844">
               −
              </span>
              <span class="mi" id="MathJax-Span-2845">
               c
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-2850">
             <span class="mrow" id="MathJax-Span-2851">
              <span class="mi" id="MathJax-Span-2852">
               ∇
              </span>
              <span class="mi" id="MathJax-Span-2853">
               f
              </span>
              <span class="mo" id="MathJax-Span-2854">
               (
              </span>
              <span class="mi" id="MathJax-Span-2855">
               x
              </span>
              <span class="mo" id="MathJax-Span-2856">
               ,
              </span>
              <span class="mi" id="MathJax-Span-2857">
               y
              </span>
              <span class="mo" id="MathJax-Span-2858">
               )
              </span>
              <span class="mo" id="MathJax-Span-2859">
               +
              </span>
              <span class="mi" id="MathJax-Span-2860">
               ν
              </span>
              <span class="mi" id="MathJax-Span-2861">
               ∇
              </span>
              <span class="mi" id="MathJax-Span-2862">
               g
              </span>
              <span class="mo" id="MathJax-Span-2863">
               (
              </span>
              <span class="mi" id="MathJax-Span-2864">
               x
              </span>
              <span class="mo" id="MathJax-Span-2865">
               ,
              </span>
              <span class="mi" id="MathJax-Span-2866">
               y
              </span>
              <span class="mo" id="MathJax-Span-2867">
               )
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-2846">
             <span class="mrow" id="MathJax-Span-2847">
              <span class="mo" id="MathJax-Span-2848">
               =
              </span>
              <span class="mn" id="MathJax-Span-2849">
               0
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-2868">
             <span class="mrow" id="MathJax-Span-2869">
              <span class="mo" id="MathJax-Span-2870">
               =
              </span>
              <span class="mn" id="MathJax-Span-2871">
               0
              </span>
             </span>
            </span>
           </span>
           <span class="mtext" id="MathJax-Span-2872">
           </span>
          </span>
         </span>
        </span>
       </div>
       <p>
        可以看到这两个条件几乎完全一样，唯一的不同就是
        <span class="editor-math-element" data-id="JTI1MjQlMjU1Q251JTI1MjQ=">
         <span class="MathJax" id="MathJax-Element-140-Frame">
          <span class="math" id="MathJax-Span-2873">
           <span class="mrow" id="MathJax-Span-2874">
            <span class="mi" id="MathJax-Span-2875">
             ν
            </span>
           </span>
          </span>
         </span>
        </span>
        的符号。由于我们并没有对
        <span class="editor-math-element" data-id="JTI1MjQlMjU1Q251JTI1MjQ=">
         <span class="MathJax" id="MathJax-Element-141-Frame">
          <span class="math" id="MathJax-Span-2876">
           <span class="mrow" id="MathJax-Span-2877">
            <span class="mi" id="MathJax-Span-2878">
             ν
            </span>
           </span>
          </span>
         </span>
        </span>
        的符号有任何限制，所以是否取负号对于这个问题没有任何影响。
       </p>
       <p>
        仍然需要提醒的是，这些条件对于一般问题只是取到最优的必要条件。但是对于大多数凸问题来说，这个条件也是充分条件。具体情况请看上面公式推导。
       </p>
       <h2>
        含有不等约束的情况
        <span id="anchor_JTI1dTU0MkIlMjV1NjcwOSUyNXU0RTBEJTI1dTdCNDklMjV1N0VBNiUyNXU2NzVGJTI1dTc2ODQlMjV1NjBDNSUyNXU1MUI1MjI=">
        </span>
       </h2>
       <p>
        上面仅仅考虑了等式约束的情况。那么含有不等式的约束情况下，
        <span class="editor-math-element" data-id="JTI1MjQlMjU1Q2xhbWJkYSUyNTI0">
         <span class="MathJax" id="MathJax-Element-142-Frame">
          <span class="math" id="MathJax-Span-2879">
           <span class="mrow" id="MathJax-Span-2880">
            <span class="mi" id="MathJax-Span-2881">
             λ
            </span>
           </span>
          </span>
         </span>
        </span>
        乘子又有什么意义呢？
       </p>
       <p>
        我们还是考虑一个和上面问题类似的问题：
       </p>
       <div class="MathJax_Display">
        <span class="MathJax" id="MathJax-Element-143-Frame">
         <span class="math" id="MathJax-Span-2882">
          <span class="mrow" id="MathJax-Span-2883">
           <span class="mtable" id="MathJax-Span-2884">
            <span class="mtd" id="MathJax-Span-2885">
             <span class="mrow" id="MathJax-Span-2886">
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-2900">
             <span class="mrow" id="MathJax-Span-2901">
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-2920">
             <span class="mrow" id="MathJax-Span-2921">
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-2887">
             <span class="mrow" id="MathJax-Span-2888">
              <span class="mtext" id="MathJax-Span-2889">
               minimize
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-2902">
             <span class="mrow" id="MathJax-Span-2903">
              <span class="mtext" id="MathJax-Span-2904">
               subject to
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-2922">
             <span class="mrow" id="MathJax-Span-2923">
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-2890">
             <span class="mrow" id="MathJax-Span-2891">
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-2905">
             <span class="mrow" id="MathJax-Span-2906">
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-2924">
             <span class="mrow" id="MathJax-Span-2925">
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-2892">
             <span class="mrow" id="MathJax-Span-2893">
              <span class="mi" id="MathJax-Span-2894">
               f
              </span>
              <span class="mo" id="MathJax-Span-2895">
               (
              </span>
              <span class="mi" id="MathJax-Span-2896">
               x
              </span>
              <span class="mo" id="MathJax-Span-2897">
               ,
              </span>
              <span class="mi" id="MathJax-Span-2898">
               y
              </span>
              <span class="mo" id="MathJax-Span-2899">
               )
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-2907">
             <span class="mrow" id="MathJax-Span-2908">
              <span class="msubsup" id="MathJax-Span-2909">
               <span class="mi" id="MathJax-Span-2910">
                g
               </span>
               <span class="mn" id="MathJax-Span-2911">
                1
               </span>
              </span>
              <span class="mo" id="MathJax-Span-2912">
               (
              </span>
              <span class="mi" id="MathJax-Span-2913">
               x
              </span>
              <span class="mo" id="MathJax-Span-2914">
               ,
              </span>
              <span class="mi" id="MathJax-Span-2915">
               y
              </span>
              <span class="mo" id="MathJax-Span-2916">
               )
              </span>
              <span class="mo" id="MathJax-Span-2917">
               ≤
              </span>
              <span class="mi" id="MathJax-Span-2918">
               c
              </span>
              <span class="mo" id="MathJax-Span-2919">
               ,
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-2926">
             <span class="mrow" id="MathJax-Span-2927">
              <span class="msubsup" id="MathJax-Span-2928">
               <span class="mi" id="MathJax-Span-2929">
                g
               </span>
               <span class="mn" id="MathJax-Span-2930">
                2
               </span>
              </span>
              <span class="mo" id="MathJax-Span-2931">
               (
              </span>
              <span class="mi" id="MathJax-Span-2932">
               x
              </span>
              <span class="mo" id="MathJax-Span-2933">
               ,
              </span>
              <span class="mi" id="MathJax-Span-2934">
               y
              </span>
              <span class="mo" id="MathJax-Span-2935">
               )
              </span>
              <span class="mo" id="MathJax-Span-2936">
               ≥
              </span>
              <span class="mi" id="MathJax-Span-2937">
               d
              </span>
             </span>
            </span>
           </span>
           <span class="mtext" id="MathJax-Span-2938">
           </span>
          </span>
         </span>
        </span>
       </div>
       <p>
        我们同样在平面内画下这个问题的图像：
        <br/>
        <a href="http://dataunion.org/wp-content/uploads/2015/01/33.png">
         <img src="http://dataunion.org/wp-content/uploads/2015/01/33.png"/>
        </a>
        <br/>
        这个和之前的图不同之处在于：约束决定的可行区域由一条直线变成了一段带状区域。这个带状区域由两条边界
        <span class="editor-math-element" data-id="JTI1MjRnJTI1Mjh4JTI1MkN5JTI1MjklMjUzRGMlMjUyNA==">
         <span class="MathJax" id="MathJax-Element-144-Frame">
          <span class="math" id="MathJax-Span-2939">
           <span class="mrow" id="MathJax-Span-2940">
            <span class="mi" id="MathJax-Span-2941">
             g
            </span>
            <span class="mo" id="MathJax-Span-2942">
             (
            </span>
            <span class="mi" id="MathJax-Span-2943">
             x
            </span>
            <span class="mo" id="MathJax-Span-2944">
             ,
            </span>
            <span class="mi" id="MathJax-Span-2945">
             y
            </span>
            <span class="mo" id="MathJax-Span-2946">
             )
            </span>
            <span class="mo" id="MathJax-Span-2947">
             =
            </span>
            <span class="mi" id="MathJax-Span-2948">
             c
            </span>
           </span>
          </span>
         </span>
        </span>
        和
        <span class="editor-math-element" data-id="JTI1MjRnJTI1Mjh4JTI1MkN5JTI1MjklMjUzRGQlMjUyNA==">
         <span class="MathJax" id="MathJax-Element-145-Frame">
          <span class="math" id="MathJax-Span-2949">
           <span class="mrow" id="MathJax-Span-2950">
            <span class="mi" id="MathJax-Span-2951">
             g
            </span>
            <span class="mo" id="MathJax-Span-2952">
             (
            </span>
            <span class="mi" id="MathJax-Span-2953">
             x
            </span>
            <span class="mo" id="MathJax-Span-2954">
             ,
            </span>
            <span class="mi" id="MathJax-Span-2955">
             y
            </span>
            <span class="mo" id="MathJax-Span-2956">
             )
            </span>
            <span class="mo" id="MathJax-Span-2957">
             =
            </span>
            <span class="mi" id="MathJax-Span-2958">
             d
            </span>
           </span>
          </span>
         </span>
        </span>
        来决定。
       </p>
       <p>
        大家立刻可以从图中发现，这个问题的最优解和之前的等式约束情况下没有任何区别。也就是依然满足条件：
       </p>
       <div class="MathJax_Display">
        <span class="MathJax" id="MathJax-Element-146-Frame">
         <span class="math" id="MathJax-Span-2959">
          <span class="mrow" id="MathJax-Span-2960">
           <span class="mtable" id="MathJax-Span-2961">
            <span class="mtd" id="MathJax-Span-2962">
             <span class="mrow" id="MathJax-Span-2963">
              <span class="msubsup" id="MathJax-Span-2964">
               <span class="mi" id="MathJax-Span-2965">
                g
               </span>
               <span class="mn" id="MathJax-Span-2966">
                1
               </span>
              </span>
              <span class="mo" id="MathJax-Span-2967">
               (
              </span>
              <span class="mi" id="MathJax-Span-2968">
               x
              </span>
              <span class="mo" id="MathJax-Span-2969">
               ,
              </span>
              <span class="mi" id="MathJax-Span-2970">
               y
              </span>
              <span class="mo" id="MathJax-Span-2971">
               )
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-2976">
             <span class="mrow" id="MathJax-Span-2977">
              <span class="mi" id="MathJax-Span-2978">
               ∇
              </span>
              <span class="mi" id="MathJax-Span-2979">
               f
              </span>
              <span class="mo" id="MathJax-Span-2980">
               (
              </span>
              <span class="mi" id="MathJax-Span-2981">
               x
              </span>
              <span class="mo" id="MathJax-Span-2982">
               ,
              </span>
              <span class="mi" id="MathJax-Span-2983">
               y
              </span>
              <span class="mo" id="MathJax-Span-2984">
               )
              </span>
              <span class="mo" id="MathJax-Span-2985">
               −
              </span>
              <span class="mi" id="MathJax-Span-2986">
               ν
              </span>
              <span class="mi" id="MathJax-Span-2987">
               ∇
              </span>
              <span class="msubsup" id="MathJax-Span-2988">
               <span class="mi" id="MathJax-Span-2989">
                g
               </span>
               <span class="mn" id="MathJax-Span-2990">
                1
               </span>
              </span>
              <span class="mo" id="MathJax-Span-2991">
               (
              </span>
              <span class="mi" id="MathJax-Span-2992">
               x
              </span>
              <span class="mo" id="MathJax-Span-2993">
               ,
              </span>
              <span class="mi" id="MathJax-Span-2994">
               y
              </span>
              <span class="mo" id="MathJax-Span-2995">
               )
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-2972">
             <span class="mrow" id="MathJax-Span-2973">
              <span class="mo" id="MathJax-Span-2974">
               =
              </span>
              <span class="mi" id="MathJax-Span-2975">
               c
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-2996">
             <span class="mrow" id="MathJax-Span-2997">
              <span class="mo" id="MathJax-Span-2998">
               =
              </span>
              <span class="mn" id="MathJax-Span-2999">
               0
              </span>
             </span>
            </span>
           </span>
           <span class="mtext" id="MathJax-Span-3000">
           </span>
          </span>
         </span>
        </span>
       </div>
       <p>
        我们来看看不等约束的KKT条件怎么说：
       </p>
       <div class="MathJax_Display">
        <span class="MathJax" id="MathJax-Element-147-Frame">
         <span class="math" id="MathJax-Span-3001">
          <span class="mrow" id="MathJax-Span-3002">
           <span class="mtable" id="MathJax-Span-3003">
            <span class="mtd" id="MathJax-Span-3004">
             <span class="mrow" id="MathJax-Span-3005">
              <span class="msubsup" id="MathJax-Span-3006">
               <span class="mi" id="MathJax-Span-3007">
                g
               </span>
               <span class="mn" id="MathJax-Span-3008">
                1
               </span>
              </span>
              <span class="mo" id="MathJax-Span-3009">
               (
              </span>
              <span class="mi" id="MathJax-Span-3010">
               x
              </span>
              <span class="mo" id="MathJax-Span-3011">
               ,
              </span>
              <span class="mi" id="MathJax-Span-3012">
               y
              </span>
              <span class="mo" id="MathJax-Span-3013">
               )
              </span>
              <span class="mo" id="MathJax-Span-3014">
               −
              </span>
              <span class="mi" id="MathJax-Span-3015">
               c
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-3020">
             <span class="mrow" id="MathJax-Span-3021">
              <span class="mi" id="MathJax-Span-3022">
               d
              </span>
              <span class="mo" id="MathJax-Span-3023">
               −
              </span>
              <span class="msubsup" id="MathJax-Span-3024">
               <span class="mi" id="MathJax-Span-3025">
                g
               </span>
               <span class="mn" id="MathJax-Span-3026">
                2
               </span>
              </span>
              <span class="mo" id="MathJax-Span-3027">
               (
              </span>
              <span class="mi" id="MathJax-Span-3028">
               x
              </span>
              <span class="mo" id="MathJax-Span-3029">
               ,
              </span>
              <span class="mi" id="MathJax-Span-3030">
               y
              </span>
              <span class="mo" id="MathJax-Span-3031">
               )
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-3036">
             <span class="mrow" id="MathJax-Span-3037">
              <span class="msubsup" id="MathJax-Span-3038">
               <span class="mi" id="MathJax-Span-3039">
                λ
               </span>
               <span class="mn" id="MathJax-Span-3040">
                1
               </span>
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-3045">
             <span class="mrow" id="MathJax-Span-3046">
              <span class="msubsup" id="MathJax-Span-3047">
               <span class="mi" id="MathJax-Span-3048">
                λ
               </span>
               <span class="mn" id="MathJax-Span-3049">
                2
               </span>
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-3054">
             <span class="mrow" id="MathJax-Span-3055">
              <span class="msubsup" id="MathJax-Span-3056">
               <span class="mi" id="MathJax-Span-3057">
                λ
               </span>
               <span class="mn" id="MathJax-Span-3058">
                1
               </span>
              </span>
              <span class="mo" id="MathJax-Span-3059">
               (
              </span>
              <span class="msubsup" id="MathJax-Span-3060">
               <span class="mi" id="MathJax-Span-3061">
                g
               </span>
               <span class="mn" id="MathJax-Span-3062">
                1
               </span>
              </span>
              <span class="mo" id="MathJax-Span-3063">
               (
              </span>
              <span class="mi" id="MathJax-Span-3064">
               x
              </span>
              <span class="mo" id="MathJax-Span-3065">
               ,
              </span>
              <span class="mi" id="MathJax-Span-3066">
               y
              </span>
              <span class="mo" id="MathJax-Span-3067">
               )
              </span>
              <span class="mo" id="MathJax-Span-3068">
               −
              </span>
              <span class="mi" id="MathJax-Span-3069">
               c
              </span>
              <span class="mo" id="MathJax-Span-3070">
               )
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-3075">
             <span class="mrow" id="MathJax-Span-3076">
              <span class="msubsup" id="MathJax-Span-3077">
               <span class="mi" id="MathJax-Span-3078">
                λ
               </span>
               <span class="mn" id="MathJax-Span-3079">
                2
               </span>
              </span>
              <span class="mo" id="MathJax-Span-3080">
               (
              </span>
              <span class="mi" id="MathJax-Span-3081">
               d
              </span>
              <span class="mo" id="MathJax-Span-3082">
               −
              </span>
              <span class="msubsup" id="MathJax-Span-3083">
               <span class="mi" id="MathJax-Span-3084">
                g
               </span>
               <span class="mn" id="MathJax-Span-3085">
                2
               </span>
              </span>
              <span class="mo" id="MathJax-Span-3086">
               (
              </span>
              <span class="mi" id="MathJax-Span-3087">
               x
              </span>
              <span class="mo" id="MathJax-Span-3088">
               ,
              </span>
              <span class="mi" id="MathJax-Span-3089">
               y
              </span>
              <span class="mo" id="MathJax-Span-3090">
               )
              </span>
              <span class="mo" id="MathJax-Span-3091">
               )
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-3096">
             <span class="mrow" id="MathJax-Span-3097">
              <span class="mi" id="MathJax-Span-3098">
               ∇
              </span>
              <span class="mi" id="MathJax-Span-3099">
               f
              </span>
              <span class="mo" id="MathJax-Span-3100">
               (
              </span>
              <span class="mi" id="MathJax-Span-3101">
               x
              </span>
              <span class="mo" id="MathJax-Span-3102">
               ,
              </span>
              <span class="mi" id="MathJax-Span-3103">
               y
              </span>
              <span class="mo" id="MathJax-Span-3104">
               )
              </span>
              <span class="mo" id="MathJax-Span-3105">
               +
              </span>
              <span class="msubsup" id="MathJax-Span-3106">
               <span class="mi" id="MathJax-Span-3107">
                λ
               </span>
               <span class="mn" id="MathJax-Span-3108">
                1
               </span>
              </span>
              <span class="mi" id="MathJax-Span-3109">
               ∇
              </span>
              <span class="msubsup" id="MathJax-Span-3110">
               <span class="mi" id="MathJax-Span-3111">
                g
               </span>
               <span class="mn" id="MathJax-Span-3112">
                1
               </span>
              </span>
              <span class="mo" id="MathJax-Span-3113">
               (
              </span>
              <span class="mi" id="MathJax-Span-3114">
               x
              </span>
              <span class="mo" id="MathJax-Span-3115">
               ,
              </span>
              <span class="mi" id="MathJax-Span-3116">
               y
              </span>
              <span class="mo" id="MathJax-Span-3117">
               )
              </span>
              <span class="mo" id="MathJax-Span-3118">
               −
              </span>
              <span class="msubsup" id="MathJax-Span-3119">
               <span class="mi" id="MathJax-Span-3120">
                λ
               </span>
               <span class="mn" id="MathJax-Span-3121">
                2
               </span>
              </span>
              <span class="mi" id="MathJax-Span-3122">
               ∇
              </span>
              <span class="msubsup" id="MathJax-Span-3123">
               <span class="mi" id="MathJax-Span-3124">
                g
               </span>
               <span class="mn" id="MathJax-Span-3125">
                2
               </span>
              </span>
              <span class="mo" id="MathJax-Span-3126">
               (
              </span>
              <span class="mi" id="MathJax-Span-3127">
               x
              </span>
              <span class="mo" id="MathJax-Span-3128">
               ,
              </span>
              <span class="mi" id="MathJax-Span-3129">
               y
              </span>
              <span class="mo" id="MathJax-Span-3130">
               )
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-3016">
             <span class="mrow" id="MathJax-Span-3017">
              <span class="mo" id="MathJax-Span-3018">
               ≤
              </span>
              <span class="mn" id="MathJax-Span-3019">
               0
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-3032">
             <span class="mrow" id="MathJax-Span-3033">
              <span class="mo" id="MathJax-Span-3034">
               ≤
              </span>
              <span class="mn" id="MathJax-Span-3035">
               0
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-3041">
             <span class="mrow" id="MathJax-Span-3042">
              <span class="mo" id="MathJax-Span-3043">
               ≥
              </span>
              <span class="mn" id="MathJax-Span-3044">
               0
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-3050">
             <span class="mrow" id="MathJax-Span-3051">
              <span class="mo" id="MathJax-Span-3052">
               ≥
              </span>
              <span class="mn" id="MathJax-Span-3053">
               0
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-3071">
             <span class="mrow" id="MathJax-Span-3072">
              <span class="mo" id="MathJax-Span-3073">
               =
              </span>
              <span class="mn" id="MathJax-Span-3074">
               0
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-3092">
             <span class="mrow" id="MathJax-Span-3093">
              <span class="mo" id="MathJax-Span-3094">
               =
              </span>
              <span class="mn" id="MathJax-Span-3095">
               0
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-3131">
             <span class="mrow" id="MathJax-Span-3132">
              <span class="mo" id="MathJax-Span-3133">
               =
              </span>
              <span class="mn" id="MathJax-Span-3134">
               0
              </span>
             </span>
            </span>
           </span>
           <span class="mtext" id="MathJax-Span-3135">
           </span>
          </span>
         </span>
        </span>
       </div>
       <p>
        这两个条件看起来一点也不像。。。。。。好吧，让我们来仔细分析一下这个KKT条件。
       </p>
       <p>
        这里的核心问题是互补松弛条件。我们上面已经说过了由于互补松弛条件的存在，
        <span class="editor-math-element" data-id="JTI1MjQlMjU1Q2xhbWJkYSUyNTI0">
         <span class="MathJax" id="MathJax-Element-148-Frame">
          <span class="math" id="MathJax-Span-3136">
           <span class="mrow" id="MathJax-Span-3137">
            <span class="mi" id="MathJax-Span-3138">
             λ
            </span>
           </span>
          </span>
         </span>
        </span>
        乘子是一个非常明确的状态指示器。如果不等约束能够取到等号，则
        <span class="editor-math-element" data-id="JTI1MjQlMjU1Q2xhbWJkYSUyNTI0">
         <span class="MathJax" id="MathJax-Element-149-Frame">
          <span class="math" id="MathJax-Span-3139">
           <span class="mrow" id="MathJax-Span-3140">
            <span class="mi" id="MathJax-Span-3141">
             λ
            </span>
           </span>
          </span>
         </span>
        </span>
        乘子是个正数；如果不等约束只能取严格不等，则
        <span class="editor-math-element" data-id="JTI1MjQlMjU1Q2xhbWJkYSUyNTI0">
         <span class="MathJax" id="MathJax-Element-150-Frame">
          <span class="math" id="MathJax-Span-3142">
           <span class="mrow" id="MathJax-Span-3143">
            <span class="mi" id="MathJax-Span-3144">
             λ
            </span>
           </span>
          </span>
         </span>
        </span>
        乘子必然为0。我们也很容易从图里看出来，要想取到最优，我们首先需要：
       </p>
       <div class="MathJax_Display">
        <span class="MathJax" id="MathJax-Element-151-Frame">
         <span class="math" id="MathJax-Span-3145">
          <span class="mrow" id="MathJax-Span-3146">
           <span class="msubsup" id="MathJax-Span-3147">
            <span class="mi" id="MathJax-Span-3148">
             g
            </span>
            <span class="mn" id="MathJax-Span-3149">
             1
            </span>
           </span>
           <span class="mo" id="MathJax-Span-3150">
            (
           </span>
           <span class="mi" id="MathJax-Span-3151">
            x
           </span>
           <span class="mo" id="MathJax-Span-3152">
            ,
           </span>
           <span class="mi" id="MathJax-Span-3153">
            y
           </span>
           <span class="mo" id="MathJax-Span-3154">
            )
           </span>
           <span class="mo" id="MathJax-Span-3155">
            −
           </span>
           <span class="mi" id="MathJax-Span-3156">
            c
           </span>
           <span class="mo" id="MathJax-Span-3157">
            =
           </span>
           <span class="mn" id="MathJax-Span-3158">
            0
           </span>
           <span class="mtext" id="MathJax-Span-3159">
           </span>
          </span>
         </span>
        </span>
       </div>
       <p>
        这时候我们有：
       </p>
       <div class="MathJax_Display">
        <span class="MathJax" id="MathJax-Element-152-Frame">
         <span class="math" id="MathJax-Span-3160">
          <span class="mrow" id="MathJax-Span-3161">
           <span class="msubsup" id="MathJax-Span-3162">
            <span class="mi" id="MathJax-Span-3163">
             λ
            </span>
            <span class="mn" id="MathJax-Span-3164">
             1
            </span>
           </span>
           <span class="mo" id="MathJax-Span-3165">
            &gt;
           </span>
           <span class="mn" id="MathJax-Span-3166">
            0
           </span>
           <span class="mo" id="MathJax-Span-3167">
            ,
           </span>
           <span class="msubsup" id="MathJax-Span-3168">
            <span class="mi" id="MathJax-Span-3169">
             λ
            </span>
            <span class="mn" id="MathJax-Span-3170">
             2
            </span>
           </span>
           <span class="mo" id="MathJax-Span-3171">
            =
           </span>
           <span class="mn" id="MathJax-Span-3172">
            0
           </span>
           <span class="mtext" id="MathJax-Span-3173">
           </span>
          </span>
         </span>
        </span>
       </div>
       <p>
        把这样两个关系带入KKT条件，可以得到：
       </p>
       <div class="MathJax_Display">
        <span class="MathJax" id="MathJax-Element-153-Frame">
         <span class="math" id="MathJax-Span-3174">
          <span class="mrow" id="MathJax-Span-3175">
           <span class="mtable" id="MathJax-Span-3176">
            <span class="mtd" id="MathJax-Span-3177">
             <span class="mrow" id="MathJax-Span-3178">
              <span class="msubsup" id="MathJax-Span-3179">
               <span class="mi" id="MathJax-Span-3180">
                g
               </span>
               <span class="mn" id="MathJax-Span-3181">
                1
               </span>
              </span>
              <span class="mo" id="MathJax-Span-3182">
               (
              </span>
              <span class="mi" id="MathJax-Span-3183">
               x
              </span>
              <span class="mo" id="MathJax-Span-3184">
               ,
              </span>
              <span class="mi" id="MathJax-Span-3185">
               y
              </span>
              <span class="mo" id="MathJax-Span-3186">
               )
              </span>
              <span class="mo" id="MathJax-Span-3187">
               −
              </span>
              <span class="mi" id="MathJax-Span-3188">
               c
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-3193">
             <span class="mrow" id="MathJax-Span-3194">
              <span class="mi" id="MathJax-Span-3195">
               ∇
              </span>
              <span class="mi" id="MathJax-Span-3196">
               f
              </span>
              <span class="mo" id="MathJax-Span-3197">
               (
              </span>
              <span class="mi" id="MathJax-Span-3198">
               x
              </span>
              <span class="mo" id="MathJax-Span-3199">
               ,
              </span>
              <span class="mi" id="MathJax-Span-3200">
               y
              </span>
              <span class="mo" id="MathJax-Span-3201">
               )
              </span>
              <span class="mo" id="MathJax-Span-3202">
               +
              </span>
              <span class="msubsup" id="MathJax-Span-3203">
               <span class="mi" id="MathJax-Span-3204">
                λ
               </span>
               <span class="mn" id="MathJax-Span-3205">
                1
               </span>
              </span>
              <span class="mi" id="MathJax-Span-3206">
               ∇
              </span>
              <span class="msubsup" id="MathJax-Span-3207">
               <span class="mi" id="MathJax-Span-3208">
                g
               </span>
               <span class="mn" id="MathJax-Span-3209">
                1
               </span>
              </span>
              <span class="mo" id="MathJax-Span-3210">
               (
              </span>
              <span class="mi" id="MathJax-Span-3211">
               x
              </span>
              <span class="mo" id="MathJax-Span-3212">
               ,
              </span>
              <span class="mi" id="MathJax-Span-3213">
               y
              </span>
              <span class="mo" id="MathJax-Span-3214">
               )
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-3189">
             <span class="mrow" id="MathJax-Span-3190">
              <span class="mo" id="MathJax-Span-3191">
               =
              </span>
              <span class="mn" id="MathJax-Span-3192">
               0
              </span>
             </span>
            </span>
            <span class="mtd" id="MathJax-Span-3215">
             <span class="mrow" id="MathJax-Span-3216">
              <span class="mo" id="MathJax-Span-3217">
               =
              </span>
              <span class="mn" id="MathJax-Span-3218">
               0
              </span>
             </span>
            </span>
           </span>
           <span class="mtext" id="MathJax-Span-3219">
           </span>
          </span>
         </span>
        </span>
       </div>
       <p>
        果然和我们之前的最优点条件一模一样了。
       </p>
       <p>
        由上面的问题可以看出来，不等约束的拉格朗日乘子
        <span class="editor-math-element" data-id="JTI1MjQlMjU1Q2xhbWJkYSUyNTI0">
         <span class="MathJax" id="MathJax-Element-154-Frame">
          <span class="math" id="MathJax-Span-3220">
           <span class="mrow" id="MathJax-Span-3221">
            <span class="mi" id="MathJax-Span-3222">
             λ
            </span>
           </span>
          </span>
         </span>
        </span>
        确实就是一个状态指示器，它起到了一个开关的作用。当不等约束能够取到等号的时候，开关打开（
        <span class="editor-math-element" data-id="JTI1MjQlMjU1Q2xhbWJkYSUyNTIwJTI1M0UlMjUyMDAlMjUyNA==">
         <span class="MathJax" id="MathJax-Element-155-Frame">
          <span class="math" id="MathJax-Span-3223">
           <span class="mrow" id="MathJax-Span-3224">
            <span class="mi" id="MathJax-Span-3225">
             λ
            </span>
            <span class="mo" id="MathJax-Span-3226">
             &gt;
            </span>
            <span class="mn" id="MathJax-Span-3227">
             0
            </span>
           </span>
          </span>
         </span>
        </span>
        ），
        <span class="editor-math-element" data-id="JTI1MjQlMjU1Q2xhbWJkYSUyNTI0">
         <span class="MathJax" id="MathJax-Element-156-Frame">
          <span class="math" id="MathJax-Span-3228">
           <span class="mrow" id="MathJax-Span-3229">
            <span class="mi" id="MathJax-Span-3230">
             λ
            </span>
           </span>
          </span>
         </span>
        </span>
        乘子的作用就和前面等式约束乘子
        <span class="editor-math-element" data-id="JTI1MjQlMjU1Q251JTI1MjQ=">
         <span class="MathJax" id="MathJax-Element-157-Frame">
          <span class="math" id="MathJax-Span-3231">
           <span class="mrow" id="MathJax-Span-3232">
            <span class="mi" id="MathJax-Span-3233">
             ν
            </span>
           </span>
          </span>
         </span>
        </span>
        的作用一样，使目标函数的梯度和约束的梯度保持线性关系；而不等约束无法取到等号的时候，开关关闭(
        <span class="editor-math-element" data-id="JTI1MjQlMjU1Q2xhbWJkYSUyNTIwJTI1M0QlMjUyMDAlMjUyOSUyNTI0">
         <span class="MathJax" id="MathJax-Element-158-Frame">
          <span class="math" id="MathJax-Span-3234">
           <span class="mrow" id="MathJax-Span-3235">
            <span class="mi" id="MathJax-Span-3236">
             λ
            </span>
            <span class="mo" id="MathJax-Span-3237">
             =
            </span>
            <span class="mn" id="MathJax-Span-3238">
             0
            </span>
            <span class="mo" id="MathJax-Span-3239">
             )
            </span>
           </span>
          </span>
         </span>
        </span>
        ，
        <span class="editor-math-element" data-id="JTI1MjQlMjU1Q2xhbWJkYSUyNTI0">
         <span class="MathJax" id="MathJax-Element-159-Frame">
          <span class="math" id="MathJax-Span-3240">
           <span class="mrow" id="MathJax-Span-3241">
            <span class="mi" id="MathJax-Span-3242">
             λ
            </span>
           </span>
          </span>
         </span>
        </span>
        乘子就带着对应的约束从梯度运算中消失了，并不去影响目标函数的梯度。
       </p>
       <h1>
        参考文献
        <span id="anchor_JTI1dTUzQzIlMjV1ODAwMyUyNXU2NTg3JTI1dTczMkUyMw==">
        </span>
       </h1>
       <ol>
        <li>
         Stephen Boyd, Lieven Vandenberghe. Convex Optimization.
        </li>
        <li>
         维基百科.
         <a href="http://zh.wikipedia.org/wiki/%E6%8B%89%E6%A0%BC%E6%9C%97%E6%97%A5%E4%B9%98%E6%95%B0">
          拉格朗日乘数
         </a>
         .
        </li>
       </ol>
       <p>
        作者：@
        <span class="username">
         wzyer
        </span>
       </p>
       <p>
        原文出处：
        <a href="http://www.moozhi.com/topic/show/54a8a261c555c08b3d59d996" target="_blank">
         http://www.moozhi.com/topic/show/54a8a261c555c08b3d59d996
        </a>
       </p>
       <p>
        <strong>
         <span style="color: #ff0000;">
          转载本文请注明原文链接以及原作者 @wzyer（新浪微博）
          <br/>
         </span>
        </strong>
       </p>
       <p>
       </p>
      </div>
      <div>
       <strong>
        注：转载文章均来自于公开网络，仅供学习使用，不会用于任何商业用途，如果侵犯到原作者的权益，请您与我们联系删除或者授权事宜，联系邮箱：contact@dataunion.org。转载数盟网站文章请注明原文章作者，否则产生的任何版权纠纷与数盟无关。
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          qrz1980 - 
						2015/10/21
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          博主你好，咨询个关于二次型的共轭函数问题，二次型f(x)=(1/2)xTQx的共轭函数是f∗(y)=(1/2)yTQ−1y，对于一般的对称矩阵Q这个结论仍成立吗？
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